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Decomposing algebraic m-isometric tuples
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Journal of Functional Analysis www.elsevier.com/locate/jfa
Decomposing algebraic m-isometric tuples Trieu Le Department of Mathematics and Statistics, The University of Toledo, Toledo, OH 43606, United States of America
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 8 June 2019 Accepted 1 December 2019 Available online xxxx Communicated by K. Seip
We show that any m-isometric tuple of commuting algebraic operators on a Hilbert space can be decomposed as a sum of a spherical isometry and a commuting nilpotent tuple. Our approach applies as well to tuples of algebraic operators that are hereditary roots of polynomials in several variables. © 2019 Elsevier Inc. All rights reserved.
MSC: 47A05 47A13 Keywords: m-isometry Nilpotent Commuting tuple
1. Introduction The notion of m-isometries was introduced and studied by Agler [3] back in the eighties. A bounded linear operator T on a complex Hilbert space H is called m-isometric if it satisfies the operator equation m
m−k
(−1)
k=0
m ∗k k T T = 0, k
E-mail address: [email protected]. https://doi.org/10.1016/j.jfa.2019.108424 0022-1236/© 2019 Elsevier Inc. All rights reserved.
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where T ∗ is the adjoint operator of T . Equivalently, for all v ∈ H, m
m−k
(−1)
k=0
m T k v2 = 0. k
In a series of papers [5–7], Agler and Stankus gave an extensive study of m-isometric operators. It is clear that any 1-isometric operator is an isometry. Multiplication by z on the Dirichlet space over the unit disk is not an isometry but it is a 2-isometry. Richter [30] showed that any cyclic 2-isometry arises from multiplication by z on certain Dirichlet-type spaces. Very recently, researchers have been interested in algebraic properties, cyclicity and supercyclicity of m-isometries, among other things. See [28,24,14,16, 15,18,13,12,26,11,22] and the references therein. It was showed by Agler, Helton and Stankus [4, Section 1.4] that any m-isometry T on a finite dimensional Hilbert space admits a decomposition T = S + N , where S is a unitary and N is a nilpotent operator satisfying SN = N S. In [12], it was showed that if S is an isometry on any Hilbert space and N is a nilpotent operator of order n commuting with S then the sum S + N is a strict (2n − 1)-isometry. This result has been generalized to m-isometries by several authors [26,11,22]. Let A be a positive operator on H. An operator T is called an (A, m)-isometry if it is a solution to the operator equation m k=0
m−k
(−1)
m ∗k T AT k = 0. k
Such operators were introduced and studied by Sid Ahmed and Saddi in [8], then by other authors [17,25,29,23,19,10]. In the case m = 1, we call such operators A-isometries. Since A is positive, the map v → vA := Av, v (where ·, · denotes the inner product on H) gives rise to a seminorm. In the case A is injective, · A becomes a norm. It follows that an operator T is (A, m)-isometric if and only if T is m-isometric with respect to ·A . As a result, several algebraic properties of (A, m)-isometries follow from the corresponding properties of m-isometries with more or less similar proofs (see [8,10]). However, there are great differences between (A, m)-isometries and m-isometries, specially when A is not injective. For example, it is known [5] that the spectrum of an m-isometry must either be a subset of the unit circle or the entire closed unit disk. On the other hand, [10, Theorem 2.3] shows that for any compact set K on the plane that intersects the unit circle, there exist a non-zero positive operator A and an (A, 1)-isometry whose spectrum is exactly K. The following question was asked in [10]. Question 1. Let T be an (A, m)-isometry on a finite dimensional Hilbert space. Is it possible to write T as a sum of an A-isometry and a commuting nilpotent operator? In this paper, we shall answer Question 1 in the affirmative. Indeed, we are able to prove a much more general result, in the setting of multivariable operator theory.
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Gleason and Richter [20] considered the multivariable setting of m-isometries and studied their properties. A commuting d-tuple of operators T = [T1 , . . . , Td ] is said to be an m-isometry if it satisfies the operator equation m
m−k
(−1)
k! α ∗ α m (T ) T = 0. α! k
(1.1)
|α|=k
k=0
Here α = (α1 , . . . , αd ) denotes a multiindex of non-negative integers. We have also used the standard multiindex notation: |α| = α1 + · · · + αd , α! = α1 ! · · · αd ! and Tα = T1α1 · · · Tdαd . Note that 1-isometric tuples are called spherical isometries. It was shown in [20] that the d-shift on the Drury-Arveson space over the unit ball in C d is d-isometric. This generalizes the single-variable fact that the unilateral shift on the Hardy space H 2 over the unit disk is an isometry. Gleason and Richter also studied spectral properties of m-isometric tuples and they constructed a list of examples of such operators, built from single-variable m-isometries. Many algebraic properties of m-isometric tuples have been discovered by the author in an unpublished work and independently by Gu [21]. As an application of our main result in this note, we shall answer the following question in the affirmative. Question 2. Let T be an m-isometric tuple acting on a finite dimensional Hilbert space. Is it possible to write T as a sum of a 1-isometric S (that is, a spherical isometry) and a nilpotent tuple N that commutes with S? To state our main result, we first generalize the notion of (A, m)-isometric operators to tuples. Let A be any bounded operator on H (we do not need to assume that A is positive). A commuting tuple T = [T1 , . . . , Td ] is said to be (A, m)-isometric if m k=0
m−k
(−1)
k! α ∗ m (T ) AT α = 0. α! k
(1.2)
|α|=k
It is clear that (I, m)-isometric tuples (here I stands for the identity operator) are the same as m-isometric tuples. We shall call (A, 1)-isometric tuples spherical A-isometric. They are tuples T that satisfies T1∗ AT1 + · · · + Td∗ ATd = A. A main result in the paper is the following theorem. Theorem 1.1. Suppose T is an (A, m)-isometric tuple on a finite dimensional Hilbert space. Then there exist a spherical A-isometric tuple S and a nilpotent tuple N commuting with S such that T = S + N.
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In the case of a single operator, Theorem 1.1 answers Question 1 in the affirmative. In the case A = I, we also obtain an affirmative answer to Question 2. 2. Hereditary calculus and applications Our approach uses a generalization of the hereditary functional calculus developed by Agler [1,2]. We begin with some definitions and notation. We use boldface lowercase letters, for example x, y, to denote d-tuples of complex variables. Let C[x, y] denote the space of polynomials in commuting variables x and y with complex coefficients. Let A be a bounded linear operator on a Hilbert space H and X, Y be two d-tuples of commuting bounded operators on H. These two tuples may not commute with each other. We denote by X∗ the tuple [X1∗ , . . . , Xd∗ ]. Let f ∈ C[x, y]. If f (x, y) = cα,β xα yβ , α,β
where the sum is finite, then we define f (A; X, Y) =
cα,β (Xα )∗ AYβ .
(2.1)
α,β
It is clear that the map f → f (A; X, Y) is linear from C[x, y] into (B(H))d . If g ∈ C[x, y] depending only on x, then g(A; X, Y) = g(X∗ )A. On the other hand, if h ∈ C[x, y] depending only on y, then h(A; X, Y) = A h(Y). Furthermore, if F = g f h, then F (A; X, Y) = g(X ∗ )f (A; X, Y)h(Y).
(2.2)
If X = Y, we shall write f (A; X) instead of f (A; X, X). In the case A = I, the identity operator, we shall use f (X, Y) to denote f (I; X, Y). Therefore, f (X) denotes f (I; X, X). We say that X is a hereditary root of f if f (X) = 0. d m Example 2.1. Define pm (x, y) = ∈ C[x, y]. It is then clear that T is j=1 xj yj − 1) m-isometric if and only if T is a hereditary root of pm , that is, pm (T) = 0. Similarly, T is (A, m)-isometric if and only if pm (A; T) = 0. Even though the map f → f (A; X, Y) is not multiplicative in general, it turns out that its kernel is an ideal of C[x, y]. This observation will play an important role in our approach. Proposition 2.2. Let A be a bounded linear operator and let X and Y be two d-tuples of commuting operators. Define J (A; X, Y) = f ∈ C[x, y] : f (A; X, Y) = 0 . Then J (A; X, Y) is an ideal of C[x, y].
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Proof. For simplicity of the notation, throughout the proof, let us write J for J (A; X, Y). It is clear that J is a vector subspace of C[x, y]. Now let f be in J and g be in C[x, y]. We need to show that gf belongs to J . By linearity, it suffices to consider the case g is a monomial g(x, y) = xα yβ for some multi-indices α and β. By (2.2), (f g)(A; X, Y) = (Xα )∗ · f (A; X, Y) · Yβ = 0, since f (A; X, Y) = 0. This shows that f g belongs to J as desired. 2 If f is a polynomial of y in the form f (y) = f¯(x) =
α α cα y ,
we define f¯(x) as
c¯α xα .
α
In the case A is positive and X = Y, we obtain an additional property of the ideal J (A; Y, Y) as follows. Proposition 2.3. Let A be a positive operator and Y be a d-tuple of commuting operators. Suppose f1 , . . . , fm are polynomials of y such that the sum f¯1 (x)f1 (y) +· · ·+ f¯m (x)fm (y) belongs to J (A; Y, Y). Then f1 (y), . . . , fm (y) also belong to J (A; Y, Y). Proof. Note that f¯j (Y∗ ) = (fj (Y))∗ for all j. By the hypotheses, we have (f1 (Y))∗ Af1 (Y) + · · · + (fm (Y))∗ Afm (Y) = 0, which implies [(A1/2 f1 (Y)]∗ [A1/2 f1 (Y)] + · · · + [A1/2 fm (Y)]∗ [A1/2 fm (Y)] = 0. It follows that for all j, we have A1/2 fj (Y) = 0, which implies Afj (Y) = 0. Therefore, fj (y) ∈ J (A; Y, Y) for all j. 2 Recall that the radical ideal of an ideal J ⊂ C[x, y], denoted by Rad(J ), is the set of all polynomials p ∈ C[x, y] such that pN ∈ J for some positive integer N . In the following proposition, we provide an interesting relation between generalized eigenvectors and eigenvalues of X and Y whenever we have f (A; X, Y) = 0. Proposition 2.4. Let X and Y be two d-tuples of commuting operators. Suppose k is a positive integer, λ = (λ1 , . . . , λd ), ω = (ω1 , . . . , ωd ) ∈ C d and u, v ∈ H such that (Xj − λj )k u = (Yj − ωj )k v = 0 for all 1 ≤ j ≤ d. Then for any polynomial f ∈ Rad(J (A; X, Y)), we have
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¯ ω)Av, u = 0. f (λ,
(2.3)
Proof. We first assume that f ∈ J (A; X, Y). Using Taylor’s expansion, we find polynomials g1 , . . . , gd and h1 , . . . , hd such that ¯ ω) − f (x, y) = f (λ,
d d ¯ j )gj (x, y) + (xj − λ hj (x, y)(yj − ωj ). j=1
j=1
Take any integer M ≥ 1 + 2d(k − 1). By the multinomial expansion, there exist polynomials G1 , . . . , Gd and H1 , . . . , Hd such that
d d
M ¯ j )k Gj (x, y) + ¯ ω) − f (x, y)) = (xj − λ Hj (x, y)(yj − ωj )k . f (λ, j=1
j=1
The left-hand side, by the binomial expansion, can be written as ¯ ω))M + f (x, y)H(x, y) (f (λ, for some polynomial H. Since f (A; X, Y) = 0, using Equation (2.2) and Proposition 2.2, we conclude that ¯ ω))M · A = (f (λ,
d d ¯ j )k Gj (A; X, Y) + (Xj∗ − λ Hj (A; X, Y)(Yj − ωj )k . j=1
j=1
Consequently, ¯ ω))M Av, u (f (λ, =
d d Gj (A; X, Y)v, (Xj − λj )k u + Hj (A; X, Y)(Yj − ωj )k v, u j=1
j=1
= 0, which implies (2.3). In the general case, there exists an integer N ≥ 1 such that f N belongs to J (A; X, Y). ¯ ω))N Av, u = 0, which again implies (2.3). This By the case we have just proved, (f (λ, completes the proof of the proposition. 2 Remark 2.5. In the case of a single operator, Proposition 2.4 provides a generalization of [4, Lemmas 18 and 19]. Our proof here is even simpler and more transparent. Question 1 and Question 2 in the introduction concern operators acting on a finite dimensional Hilbert space. It turns out that this condition can be replaced by a weaker
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one. Recall that a linear operator T is called algebraic if there exist complex constants c0 , c1 , . . . , c such that c0 I + c1 T + · · · + c T = 0. Algebraic operator roots of polynomials were investigated in [4]. We first discuss some preparatory results on algebraic operators acting on a general complex vector space V. It is well known that if T is an algebraic linear operator on V, then the spectrum σ(T ) is finite and there exists a direct sum decomposition V = ⊕a∈σ(T ) Va , where each Va is an invariant subspace for T (the subspace Va is a closed subspace if V is a normed space and T is bounded) and T −aI is nilpotent on Va . Indeed, if the minimal polynomial of T is factored in the form p(z) = (z − a1 )m1 · · · (z − a )m , where a1 , . . . , a are pairwise distinct and m1 , . . . , m ≥ 1, then σ(T ) = {a1 , . . . , a } and Vaj = ker(T − aj )mj for 1 ≤ j ≤ . See, for example, [32, Section 6.3], which discusses operators acting on finite dimensional vector spaces. However, the arguments apply to algebraic operators on infinite dimensional vector spaces as well. Suppose now T = [T1 , . . . , Td ] is a tuple of commuting algebraic operators on V. We first decompose V as above with respect to the spectrum σ(T1 ). Since each subspace in the decomposition is invariant for all Tj , we again decompose such subspace with respect to the spectrum σ(T2 ). Continuing this process, we obtain a finite set Λ ⊂ C d and a direct sum decomposition V = ⊕λ∈Λ Vλ such that for each λ = (λ1 , . . . , λd ) ∈ Λ and 1 ≤ j ≤ d, the subspace Vλ is invariant for T and Tj − λj I is nilpotent on Vλ . Let Eλ denote the canonical projection (possibly non-orthogonal) from V onto Vλ . Then we have 2 λ∈Λ Eλ = I, Eλ = Eλ , and Eλ Eγ = 0 if λ = γ. Define S=
λ∈Λ
λ · Eλ =
λ1 Eλ , . . . ,
λ∈Λ
λ d Eλ
(2.4)
λ∈Λ
Then S is a tuple of commuting operators which commutes with T, and T−S is nilpotent. For any multiindex α, we have Sα = S1α1 · · · Sdαd =
λα Eλ .
λ∈Λ
In the case V is a normed space and T is bounded, each operator in the tuple S is bounded as well. We now prove a very general result, which will provide affirmative answers to Questions 1 and 2 in the introduction.
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Theorem 2.6. Let X and Y be two d-tuples of commuting algebraic operators on a Hilbert space H. Let U (respectively, V) be the commuting tuple associated with X (respectively, Y) as in (2.4). Then Rad(J (A; X, Y)) ⊆ J (A; U, V).
(2.5)
Proof. Write X = [X1 , . . . , Xd ] and decompose H = ⊕λ∈Λ Hλ such that for each λ = (λ1 , . . . , λd ) ∈ Λ, the subspace Hλ is invariant for X and Xj − λj I is nilpotent on Hλ . Let Uλ denote the canonical projection from H onto Hλ . Then U = λ∈Λ λ · Uλ and for any multiindex α, we have Uα =
λα · Uλ .
λ∈Λ
Similarly, write Y = [Y1 , . . . , Yd ] and decompose H = ⊕ω∈Ω Kω . Let Vω be the canonical projection from H onto Kω . Then V = ω∈Ω ω · Vω and for any multiindex β, Vβ =
ω β · Vω .
ω∈Ω
Take any polynomial p ∈ Rad(J (A; X, Y)). For λ ∈ Λ, ω ∈ Ω and vectors u ∈ Hλ and v ∈ Kω , there exists an integer k ≥ 1 sufficiently large such that (Xj − λj I)k u = (Yj − ωj I)k v = 0 ¯ ω)Av, u = 0, which implies for all 1 ≤ j ≤ d. Proposition 2.4 shows that p(λ, ¯ ω)U ∗ AVω = 0. p(λ, λ Write p(x, y) =
α,β cα,β x
α β
y . We compute
p(A; U, V) =
cα,β U∗α AVβ
α,β
=
cα,β
α,β
=
λ∈Λ,ω∈Ω
=
λ∈Λ
β ¯αU ∗ A ω · V λ ω λ ω∈Ω
¯ α ω β U ∗ AVω cα,β λ λ
α,β
¯ ω)U ∗ AVω = 0. p(λ, λ
λ∈Λ,ω∈Ω
We conclude that p ∈ J (A; U, V). Since p ∈ Rad(J (A; X, Y)) was arbitrary, the proof of the theorem is complete. 2
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Theorem 2.6 enjoys numerous interesting applications that we now describe. Proof of Theorem 1.1. We shall prove the theorem under a more general assumption that T is a tuple of commuting algebraic operators. Since T is (A, m)-isometric, the polynod mial ( j=1 xj yj − 1)m belongs to the ideal J (A; T, T). It follows that the polynomial d p(x, y) = j=1 xj yj − 1 belongs to the radical ideal of J (A; T, T). By Theorem 2.6, we may decompose T = S + N, where N is a nilpotent tuple commuting with S and p(A; S, S) = 0, which means that S is a spherical A-isometry. 2 Example 2.7. Recall that an operator T is called (m, n)-isosymmetric (see [33]) if T is a hereditary root of f (x, y) = (xy − 1)m (x − y)n . Theorem 2.6 shows that any such algebraic T can be decomposed as T = S + N , where N is nilpotent, S is isosymmetric (i.e. (1, 1)-isosymmetric) and SN = N S. Example 2.8. Several researchers [27,9] have investigated the so-called toral m-isometric tuples. It is straightforward to generalize this notion to toral (A, m)-isometric tuples, which are commuting d-tuples T that satisfy |α| m1 , . . . , md (Tα )∗ A Tα = 0. (−1) α 0≤α1 ≤m1 ... 0≤αd ≤md
for all m1 +· · ·+md = m. Equivalently, T is a common hereditary root of all polynomials of the form (1 − x1 y1 )m1 · · · (1 − xd yd )md for m1 + · · · + md = m. This means that all these polynomials belong to the ideal J (A; T, T). We see that toral (A, 1)-isometries are just commuting tuples T such that each Tj is an A-isometry, that is, Tj∗ ATj = A. Note that for any toral (A, m)-isometry T, the radical ideal Rad(J (A; T, T)) contains all the polynomials 1 − xj yj : j = 1, 2, . . . , d . Theorem 2.6 asserts that T = S + N, where S is a toral (A, 1)-isometry and N is a nilpotent tuple commuting with S. 3. On 2-isometric tuples It is well known that any 2-isometry on a finite dimensional Hilbert space must actually be an isometry. On the other hand, there are many examples of finite dimensional 2-isometric tuples that are not spherical isometries. The following class of examples is given in Richter’s talk [31]. Example 3.1. If α = (α1 , . . . , αd ) ∈ ∂Bd and Vj : C m → C n such that then W = (W1 , . . . , Wd ) with Wj = defines a 2-isometric d-tuple.
αj In 0
Vj αj Im
d j=1
αj Vj = 0,
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The following result was stated in [31] without a proof and as far as the author is aware of, it has not appeared in a published paper. Theorem 3.2 (Richter-Sundberg). If T is a 2-isometric tuple on a finite dimensional space, then T = U ⊕ W, where U is a spherical unitary and W is a direct sum of operator tuples unitarily equivalent to those in Example 3.1. In this section, we shall assume that A is self-adjoint and investigate (A, 2)-isometric d-tuples. We obtain a characterization for such tuples that generalizes the above theorem. We first provide a generalization of Example 3.1. We call N = (N1 , . . . , Nd ) an (A, n)-nilpotent tuple if ANα = 0 for any indices α with |α| = n. Proposition 3.3. Assume that A is a self-adjoint operator. Let S be an (A, 1)-isometry and N an (A, 2)-nilpotent tuple such that S commutes with N. Suppose S1∗ AN1 + · · · + Sd∗ ANd = 0, then S + N is an (A, 2)-isometry. Proof. By the assumption, we have ANj Nk = Nj∗ Nk∗ A = 0 for 1 ≤ j, k ≤ d, d d d ∗ ∗ ∗ j=1 Sj ASj = A, and j=1 Sj ANj = j=1 Nj ASj = 0. It follows that d d (Sj + Nj )∗ A (Sj + Nj ) = A + Nj∗ ANj . j=1
j=1
We then compute
(Sk + Nk )∗ (Sj + Nj )∗ A (Sj + Nj )(Sk + Nk )
1≤k,j≤d
=
d
(Sk∗
+
Nk∗ )(A
+
d
Nj∗ ANj )(Sk + Nk )
j=1
k=1
=
d
(Sk∗ + Nk∗ )A (Sk + Nk ) +
k=1
=A+
(Sk∗ + Nk∗ )Nj∗ ANj (Sk + Nk )
1≤k,j≤d d
Nk∗ ANk +
k=1
=A+
d k=1
Sk∗ Nj∗ ANj Sk
1≤k,j≤d
Nk∗ ANk +
d j=1
Nj∗
d k=1
Sk∗ ASk Nj
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=A+
d
Nk∗ ANk +
d
Nj∗ ANj
j=1
k=1
=A+2
d
11
Nj∗ ANj
j=1
=2
d (Sj + Nj )∗ A (Sj + Nj ) − A. j=1
Consequently, the sum S + N is an (A, 2)-isometric tuple. 2 Remark 3.4. We have provided a direct proof of Proposition 2.3. Using the hereditary functional calculus and the approach in [26], one may generalize the result to the case S being an (A, m)-isometry and N an (A, n)-nilpotent commuting with S. Under such an assumption, if S1∗ AN1 + · · · + Sd∗ ANd = 0, then S + N is an (A, m + 2n − 3)-isometry. We leave the details for the interested reader. We now show that any algebraic (A, 2)-isometric tuple has the form given in Proposition 3.3 and as a result, provide a proof of Richter-Sundberg’s theorem. Theorem 3.5. Assume that A is a positive operator. Let T be an algebraic (A, 2)-isometric tuple on H. Then there exists an (A, 1)-isometric tuple S and a tuple N commuting with d S such that T = S + N, =1 S∗ AN = 0, and ANj N = 0 for all 1 ≤ j, ≤ d (we call such N an (A, 2)-nilpotent tuple). In the case H is finite dimensional and A = I, the identity operator, we recover Theorem 3.2. Proof. Recall that there exists a finite set Λ ⊂ C d and a direct sum decomposition H = ⊕λ∈Λ Hλ such that for each λ ∈ Λ, the subspace Hλ is invariant for T and Tj − λj I is nilpotent on Hλ . Let S be defined as in (2.4) and put N = T−S. From the construction, N is nilpotent and Theorem 2.6 shows that S is (A, 1)-isometric. We shall show that N satisfies the required properties. Restricting on each invariant subspace Hλ , we only need to consider the case H = Hλ and so S = λI. Proposition 2.4 asserts that (|λ|2 − 1)Av, u = 0 for all v, u ∈ H. If |λ| = 1, then A = 0 and the conclusion follows. Now we assume that |λ| = 1. Since N is nilpotent, there exists a positive integer r such that A Nα = 0 whenever |α| = r. We claim that r may be taken to be 2. To prove the claim, we assume r ≥ 3 and show that A Nα = 0 for all |α| = r − 1. Since T = λI + N is (A, 2)-isometric, the tuple N is an A-root of the polynomial
p(x, y) =
d d
2
2 ¯ j )(yj + λj ) − 1 = ¯ j yj . (xj + λ xj yj + λj xj + λ j=1
j=1
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On the other hand, N is an A-root of xα and yα for all |α| = r. This shows that p(x, y), xα and yα belong to J (A; N, N) for all |α| = r. To simplify the notation, we shall denote J (A; N, N) by J in the rest of the proof. Take any multiindex β with |β| = r − 2. We write xβ p(x, y)yβ = xβ
d
λj x j
d
j=1
¯ y yβ + xγ Hγ (x, y) + Gγ (x, y) yγ λ |γ|≥r
=1
for some polynomials Hγ and Gγ . Since the left-hand side and the second term on the right-hand side belong to J , which is an ideal, we conclude that xβ
d
λj x j
d
j=1
¯ y yβ ∈ J . λ
=1
d ¯ y yβ and xβ d λj xj are in J . Now Proposition 2.3 shows that both λ =1 j=1 for any multiindex γ with |γ| = r − 3, we compute xγ p(x, y)yγ = xγ
d
d
2 xj yj yγ + xβ λj xj Pβ (x, y) |β|=r−2
j=1
+
d |β|=r−2
j=1
λj xj yβ Qβ (x, y).
j=1
Since the left-hand side and the last two sums on the right-hand side belong to J , it d follows that xγ ( j=1 xj yj )2 yγ belongs to J . Another application of Proposition 2.3 then shows that yj y yγ belongs to J for all 1 ≤ j, ≤ d. That is, yα belongs to J whenever |α| = r − 1 (as long as r ≥ 3). As a consequence, we see that yα , and hence xα , belong to J for all |α| = 2. This together with the fact that p(x, y) ∈ J forces d d ¯ d ¯ ( j=1 λj xj )( =1 λ y ) to belong to J , which implies that =1 λ y is in J . We have d ¯ d ∗ then shown ANj N = 0 for all 1 ≤ j, ≤ d and S AN = λ AN = 0, as =1
=1
desired. Now let us consider T a 2-isometric tuple on a finite dimensional space H. Recall that we have the decomposition H = ⊕λ∈Λ Hλ such that for each λ ∈ Λ, the subspace Hλ is invariant for T and Tj − λj I is nilpotent on Hλ . By Proposition 2.4, we have (ω, λ − 1)2 v, u = 0 for all v ∈ Hω and u ∈ Hλ . It follows that |λ| = 1 for all λ ∈ Λ and Hλ ⊥ Hω whenever λ = ω. As a result, each subspace Hλ is reducing for T. To complete the proof, it suffices to consider H = Hλ . We shall show that either T is a spherical unitary or it is unitarily equivalent to a tuple given in Example 3.1. Indeed, d ¯ we have T = λI + N, where =1 λ N = 0 and Nj N = 0 for all 1 ≤ j, ≤ d. If N = 0, then T is a spherical unitary. Otherwise, let M = ker(N1 ) ∩ · · · ∩ ker(Nd ). Then N (H) ⊆ M for all 1 ≤ ≤ d. As a consequence, with respect to the orthogonal decomposition H = M ⊕ M⊥ , each N has the form
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N =
0 V 0 0
13
d ¯ d ¯ for some V : M⊥ → M. Since =1 λ N = 0, we have =1 λ V = 0. It follows that T is unitarily equivalent to an operator tuple in Example 3.1. 2 Acknowledgments The author wishes to thank the referee for helpful suggestions which improved the presentation of the paper. References [1] J. Agler, Subjordan operators, PhD Thesis, ProQuest LLC, Indiana University, Ann Arbor, MI, 1980. [2] J. Agler, Sub-Jordan operators: Bishop’s theorem, spectral inclusion, and spectral sets, J. Oper. Theory 7 (2) (1982) 373–395. [3] J. Agler, A disconjugacy theorem for Toeplitz operators, Am. J. Math. 112 (1) (1990) 1–14. [4] J. Agler, J.W. Helton, M. Stankus, Classification of hereditary matrices, Linear Algebra Appl. 274 (1998) 125–160. [5] J. Agler, M. Stankus, m-isometric transformations of Hilbert space I, Integral Equ. Oper. Theory 21 (4) (1995) 383–429. [6] J. Agler, M. Stankus, m-isometric transformations of Hilbert space II, Integral Equ. Oper. Theory 23 (1) (1995) 1–48. [7] J. Agler, M. Stankus, m-isometric transformations of Hilbert space III, Integral Equ. Oper. Theory 24 (4) (1996) 379–421. [8] O.A.M. Sid Ahmed, A. Saddi, A-m-isometric operators in semi-Hilbertian spaces, Linear Algebra Appl. 436 (10) (2012) 3930–3942. [9] A. Athavale, V. Sholapurkar, Completely hyperexpansive operator tuples, Positivity 3 (3) (1999) 245–257. [10] T. Bermúdez, A. Saddi, H. Zaway, (A, m)-isometries on Hilbert spaces, Linear Algebra Appl. 540 (2018) 95–111. [11] T. Bermúdez, A. Martinón, V. Müller, J.A. Noda, Perturbation of m-Isometries by nilpotent operators, Abstr. Appl. Anal. (2014) 745479. [12] T. Bermúdez, A. Martinón, J.A. Noda, An isometry plus a nilpotent operator is an m-isometry. Applications, J. Math. Anal. Appl. 407 (2) (2013) 505–512. [13] T. Bermúdez, A. Martinón, J.A. Noda, Products of m-isometries, Linear Algebra Appl. 438 (1) (2013) 80–86. [14] F. Botelho, On the existence of n-isometries on p spaces, Acta Sci. Math. (Szeged) 76 (1–2) (2010) 183–192. [15] M. Ch¯ o, S. Ôta, K. Tanahashi, Invertible weighted shift operators which are m-isometries, Proc. Am. Math. Soc. 141 (12) (2013) 4241–4247. [16] B.P. Duggal, Tensor product of n-isometries, Linear Algebra Appl. 437 (1) (2012) 307–318. [17] B.P. Duggal, Tensor product of n-isometries III, Funct. Anal. Approx. Comput. 4 (2) (2012) 61–67. [18] M. Faghih-Ahmadi, K. Hedayatian, m-isometric weighted shifts and reflexivity of some operators, Rocky Mt. J. Math. 43 (1) (2013) 123–133. [19] M. Faghih-Ahmadi, Powers of A-m-isometric operators and their supercyclicity, Bull. Malays. Math. Sci. Soc. 39 (3) (2016) 901–911. [20] J. Gleason, S. Richter, m-isometric commuting tuples of operators on a Hilbert space, Integral Equ. Oper. Theory 56 (2) (2006) 181–196. [21] C. Gu, Examples of m-isometric tuples of operators on a Hilbert space, J. Korean Math. Soc. 55 (1) (2018) 225–251. [22] C. Gu, M. Stankus, Some results on higher order isometries and symmetries: products and sums with a nilpotent operator, Linear Algebra Appl. 469 (2015) 500–509.
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[23] K. Hedayatian, N -supercyclicity of an A-m-isometry, Honam Math. J. 37 (3) (2015) 281–285. [24] Z.J. Jabłoński, Complete hyperexpansivity, subnormality and inverted boundedness conditions, Integral Equ. Oper. Theory 44 (3) (2002) 316–336. [25] S. Jung, Y. Kim, E. Ko, J.E. Lee, On (A, m)-expansive operators, Stud. Math. 213 (1) (2012) 3–23. [26] T. Le, Algebraic properties of operator roots of polynomials, J. Math. Anal. Appl. 421 (2) (2015) 1238–1246. [27] A. Mohammadi-Moghaddam, K. Hedayatian, On the dynamics of the d-tuples of m-isometries, Rocky Mt. J. Math. 49 (1) (2019) 283–305. [28] S.M. Patel, 2-isometric operators, Glas. Mat. Ser. III 37 (1) (2002) 141–145. [29] R. Rabaoui, A. Saddi, On the orbit of an A-m-isometry, Ann. Math. Sil. 2012 (26) (2013) 75–91. [30] S. Richter, A representation theorem for cyclic analytic two-isometries, Trans. Am. Math. Soc. 328 (1) (1991) 325–349. [31] S. Richter, A model for two-isometric operator tuples with finite defect, https://www.math.utk. edu/~richter/talk/Cornell_Sept_2011, 2011. [32] G.E. Shilov, Linear Algebra, english ed., Dover Publications, Inc., New York, 1977, translated from the Russian and edited by Richard A. Silverman. [33] M. Stankus, m-isometries, n-symmetries and other linear transformations which are hereditary roots, Integral Equ. Oper. Theory 75 (3) (2013) 301–321.
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Contents lists available at ScienceDirect
Journal of Functional Analysis www.elsevier.com/locate/jfa
Decomposing algebraic m-isometric tuples Trieu Le Department of Mathematics and Statistics, The University of Toledo, Toledo, OH 43606, United States of America
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 8 June 2019 Accepted 1 December 2019 Available online xxxx Communicated by K. Seip
We show that any m-isometric tuple of commuting algebraic operators on a Hilbert space can be decomposed as a sum of a spherical isometry and a commuting nilpotent tuple. Our approach applies as well to tuples of algebraic operators that are hereditary roots of polynomials in several variables. © 2019 Elsevier Inc. All rights reserved.
MSC: 47A05 47A13 Keywords: m-isometry Nilpotent Commuting tuple
1. Introduction The notion of m-isometries was introduced and studied by Agler [3] back in the eighties. A bounded linear operator T on a complex Hilbert space H is called m-isometric if it satisfies the operator equation m
m−k
(−1)
k=0
m ∗k k T T = 0, k
E-mail address: [email protected]. https://doi.org/10.1016/j.jfa.2019.108424 0022-1236/© 2019 Elsevier Inc. All rights reserved.
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where T ∗ is the adjoint operator of T . Equivalently, for all v ∈ H, m
m−k
(−1)
k=0
m T k v2 = 0. k
In a series of papers [5–7], Agler and Stankus gave an extensive study of m-isometric operators. It is clear that any 1-isometric operator is an isometry. Multiplication by z on the Dirichlet space over the unit disk is not an isometry but it is a 2-isometry. Richter [30] showed that any cyclic 2-isometry arises from multiplication by z on certain Dirichlet-type spaces. Very recently, researchers have been interested in algebraic properties, cyclicity and supercyclicity of m-isometries, among other things. See [28,24,14,16, 15,18,13,12,26,11,22] and the references therein. It was showed by Agler, Helton and Stankus [4, Section 1.4] that any m-isometry T on a finite dimensional Hilbert space admits a decomposition T = S + N , where S is a unitary and N is a nilpotent operator satisfying SN = N S. In [12], it was showed that if S is an isometry on any Hilbert space and N is a nilpotent operator of order n commuting with S then the sum S + N is a strict (2n − 1)-isometry. This result has been generalized to m-isometries by several authors [26,11,22]. Let A be a positive operator on H. An operator T is called an (A, m)-isometry if it is a solution to the operator equation m k=0
m−k
(−1)
m ∗k T AT k = 0. k
Such operators were introduced and studied by Sid Ahmed and Saddi in [8], then by other authors [17,25,29,23,19,10]. In the case m = 1, we call such operators A-isometries. Since A is positive, the map v → vA := Av, v (where ·, · denotes the inner product on H) gives rise to a seminorm. In the case A is injective, · A becomes a norm. It follows that an operator T is (A, m)-isometric if and only if T is m-isometric with respect to ·A . As a result, several algebraic properties of (A, m)-isometries follow from the corresponding properties of m-isometries with more or less similar proofs (see [8,10]). However, there are great differences between (A, m)-isometries and m-isometries, specially when A is not injective. For example, it is known [5] that the spectrum of an m-isometry must either be a subset of the unit circle or the entire closed unit disk. On the other hand, [10, Theorem 2.3] shows that for any compact set K on the plane that intersects the unit circle, there exist a non-zero positive operator A and an (A, 1)-isometry whose spectrum is exactly K. The following question was asked in [10]. Question 1. Let T be an (A, m)-isometry on a finite dimensional Hilbert space. Is it possible to write T as a sum of an A-isometry and a commuting nilpotent operator? In this paper, we shall answer Question 1 in the affirmative. Indeed, we are able to prove a much more general result, in the setting of multivariable operator theory.
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Gleason and Richter [20] considered the multivariable setting of m-isometries and studied their properties. A commuting d-tuple of operators T = [T1 , . . . , Td ] is said to be an m-isometry if it satisfies the operator equation m
m−k
(−1)
k! α ∗ α m (T ) T = 0. α! k
(1.1)
|α|=k
k=0
Here α = (α1 , . . . , αd ) denotes a multiindex of non-negative integers. We have also used the standard multiindex notation: |α| = α1 + · · · + αd , α! = α1 ! · · · αd ! and Tα = T1α1 · · · Tdαd . Note that 1-isometric tuples are called spherical isometries. It was shown in [20] that the d-shift on the Drury-Arveson space over the unit ball in C d is d-isometric. This generalizes the single-variable fact that the unilateral shift on the Hardy space H 2 over the unit disk is an isometry. Gleason and Richter also studied spectral properties of m-isometric tuples and they constructed a list of examples of such operators, built from single-variable m-isometries. Many algebraic properties of m-isometric tuples have been discovered by the author in an unpublished work and independently by Gu [21]. As an application of our main result in this note, we shall answer the following question in the affirmative. Question 2. Let T be an m-isometric tuple acting on a finite dimensional Hilbert space. Is it possible to write T as a sum of a 1-isometric S (that is, a spherical isometry) and a nilpotent tuple N that commutes with S? To state our main result, we first generalize the notion of (A, m)-isometric operators to tuples. Let A be any bounded operator on H (we do not need to assume that A is positive). A commuting tuple T = [T1 , . . . , Td ] is said to be (A, m)-isometric if m k=0
m−k
(−1)
k! α ∗ m (T ) AT α = 0. α! k
(1.2)
|α|=k
It is clear that (I, m)-isometric tuples (here I stands for the identity operator) are the same as m-isometric tuples. We shall call (A, 1)-isometric tuples spherical A-isometric. They are tuples T that satisfies T1∗ AT1 + · · · + Td∗ ATd = A. A main result in the paper is the following theorem. Theorem 1.1. Suppose T is an (A, m)-isometric tuple on a finite dimensional Hilbert space. Then there exist a spherical A-isometric tuple S and a nilpotent tuple N commuting with S such that T = S + N.
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In the case of a single operator, Theorem 1.1 answers Question 1 in the affirmative. In the case A = I, we also obtain an affirmative answer to Question 2. 2. Hereditary calculus and applications Our approach uses a generalization of the hereditary functional calculus developed by Agler [1,2]. We begin with some definitions and notation. We use boldface lowercase letters, for example x, y, to denote d-tuples of complex variables. Let C[x, y] denote the space of polynomials in commuting variables x and y with complex coefficients. Let A be a bounded linear operator on a Hilbert space H and X, Y be two d-tuples of commuting bounded operators on H. These two tuples may not commute with each other. We denote by X∗ the tuple [X1∗ , . . . , Xd∗ ]. Let f ∈ C[x, y]. If f (x, y) = cα,β xα yβ , α,β
where the sum is finite, then we define f (A; X, Y) =
cα,β (Xα )∗ AYβ .
(2.1)
α,β
It is clear that the map f → f (A; X, Y) is linear from C[x, y] into (B(H))d . If g ∈ C[x, y] depending only on x, then g(A; X, Y) = g(X∗ )A. On the other hand, if h ∈ C[x, y] depending only on y, then h(A; X, Y) = A h(Y). Furthermore, if F = g f h, then F (A; X, Y) = g(X ∗ )f (A; X, Y)h(Y).
(2.2)
If X = Y, we shall write f (A; X) instead of f (A; X, X). In the case A = I, the identity operator, we shall use f (X, Y) to denote f (I; X, Y). Therefore, f (X) denotes f (I; X, X). We say that X is a hereditary root of f if f (X) = 0. d m Example 2.1. Define pm (x, y) = ∈ C[x, y]. It is then clear that T is j=1 xj yj − 1) m-isometric if and only if T is a hereditary root of pm , that is, pm (T) = 0. Similarly, T is (A, m)-isometric if and only if pm (A; T) = 0. Even though the map f → f (A; X, Y) is not multiplicative in general, it turns out that its kernel is an ideal of C[x, y]. This observation will play an important role in our approach. Proposition 2.2. Let A be a bounded linear operator and let X and Y be two d-tuples of commuting operators. Define J (A; X, Y) = f ∈ C[x, y] : f (A; X, Y) = 0 . Then J (A; X, Y) is an ideal of C[x, y].
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Proof. For simplicity of the notation, throughout the proof, let us write J for J (A; X, Y). It is clear that J is a vector subspace of C[x, y]. Now let f be in J and g be in C[x, y]. We need to show that gf belongs to J . By linearity, it suffices to consider the case g is a monomial g(x, y) = xα yβ for some multi-indices α and β. By (2.2), (f g)(A; X, Y) = (Xα )∗ · f (A; X, Y) · Yβ = 0, since f (A; X, Y) = 0. This shows that f g belongs to J as desired. 2 If f is a polynomial of y in the form f (y) = f¯(x) =
α α cα y ,
we define f¯(x) as
c¯α xα .
α
In the case A is positive and X = Y, we obtain an additional property of the ideal J (A; Y, Y) as follows. Proposition 2.3. Let A be a positive operator and Y be a d-tuple of commuting operators. Suppose f1 , . . . , fm are polynomials of y such that the sum f¯1 (x)f1 (y) +· · ·+ f¯m (x)fm (y) belongs to J (A; Y, Y). Then f1 (y), . . . , fm (y) also belong to J (A; Y, Y). Proof. Note that f¯j (Y∗ ) = (fj (Y))∗ for all j. By the hypotheses, we have (f1 (Y))∗ Af1 (Y) + · · · + (fm (Y))∗ Afm (Y) = 0, which implies [(A1/2 f1 (Y)]∗ [A1/2 f1 (Y)] + · · · + [A1/2 fm (Y)]∗ [A1/2 fm (Y)] = 0. It follows that for all j, we have A1/2 fj (Y) = 0, which implies Afj (Y) = 0. Therefore, fj (y) ∈ J (A; Y, Y) for all j. 2 Recall that the radical ideal of an ideal J ⊂ C[x, y], denoted by Rad(J ), is the set of all polynomials p ∈ C[x, y] such that pN ∈ J for some positive integer N . In the following proposition, we provide an interesting relation between generalized eigenvectors and eigenvalues of X and Y whenever we have f (A; X, Y) = 0. Proposition 2.4. Let X and Y be two d-tuples of commuting operators. Suppose k is a positive integer, λ = (λ1 , . . . , λd ), ω = (ω1 , . . . , ωd ) ∈ C d and u, v ∈ H such that (Xj − λj )k u = (Yj − ωj )k v = 0 for all 1 ≤ j ≤ d. Then for any polynomial f ∈ Rad(J (A; X, Y)), we have
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¯ ω)Av, u = 0. f (λ,
(2.3)
Proof. We first assume that f ∈ J (A; X, Y). Using Taylor’s expansion, we find polynomials g1 , . . . , gd and h1 , . . . , hd such that ¯ ω) − f (x, y) = f (λ,
d d ¯ j )gj (x, y) + (xj − λ hj (x, y)(yj − ωj ). j=1
j=1
Take any integer M ≥ 1 + 2d(k − 1). By the multinomial expansion, there exist polynomials G1 , . . . , Gd and H1 , . . . , Hd such that
d d
M ¯ j )k Gj (x, y) + ¯ ω) − f (x, y)) = (xj − λ Hj (x, y)(yj − ωj )k . f (λ, j=1
j=1
The left-hand side, by the binomial expansion, can be written as ¯ ω))M + f (x, y)H(x, y) (f (λ, for some polynomial H. Since f (A; X, Y) = 0, using Equation (2.2) and Proposition 2.2, we conclude that ¯ ω))M · A = (f (λ,
d d ¯ j )k Gj (A; X, Y) + (Xj∗ − λ Hj (A; X, Y)(Yj − ωj )k . j=1
j=1
Consequently, ¯ ω))M Av, u (f (λ, =
d d Gj (A; X, Y)v, (Xj − λj )k u + Hj (A; X, Y)(Yj − ωj )k v, u j=1
j=1
= 0, which implies (2.3). In the general case, there exists an integer N ≥ 1 such that f N belongs to J (A; X, Y). ¯ ω))N Av, u = 0, which again implies (2.3). This By the case we have just proved, (f (λ, completes the proof of the proposition. 2 Remark 2.5. In the case of a single operator, Proposition 2.4 provides a generalization of [4, Lemmas 18 and 19]. Our proof here is even simpler and more transparent. Question 1 and Question 2 in the introduction concern operators acting on a finite dimensional Hilbert space. It turns out that this condition can be replaced by a weaker
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one. Recall that a linear operator T is called algebraic if there exist complex constants c0 , c1 , . . . , c such that c0 I + c1 T + · · · + c T = 0. Algebraic operator roots of polynomials were investigated in [4]. We first discuss some preparatory results on algebraic operators acting on a general complex vector space V. It is well known that if T is an algebraic linear operator on V, then the spectrum σ(T ) is finite and there exists a direct sum decomposition V = ⊕a∈σ(T ) Va , where each Va is an invariant subspace for T (the subspace Va is a closed subspace if V is a normed space and T is bounded) and T −aI is nilpotent on Va . Indeed, if the minimal polynomial of T is factored in the form p(z) = (z − a1 )m1 · · · (z − a )m , where a1 , . . . , a are pairwise distinct and m1 , . . . , m ≥ 1, then σ(T ) = {a1 , . . . , a } and Vaj = ker(T − aj )mj for 1 ≤ j ≤ . See, for example, [32, Section 6.3], which discusses operators acting on finite dimensional vector spaces. However, the arguments apply to algebraic operators on infinite dimensional vector spaces as well. Suppose now T = [T1 , . . . , Td ] is a tuple of commuting algebraic operators on V. We first decompose V as above with respect to the spectrum σ(T1 ). Since each subspace in the decomposition is invariant for all Tj , we again decompose such subspace with respect to the spectrum σ(T2 ). Continuing this process, we obtain a finite set Λ ⊂ C d and a direct sum decomposition V = ⊕λ∈Λ Vλ such that for each λ = (λ1 , . . . , λd ) ∈ Λ and 1 ≤ j ≤ d, the subspace Vλ is invariant for T and Tj − λj I is nilpotent on Vλ . Let Eλ denote the canonical projection (possibly non-orthogonal) from V onto Vλ . Then we have 2 λ∈Λ Eλ = I, Eλ = Eλ , and Eλ Eγ = 0 if λ = γ. Define S=
λ∈Λ
λ · Eλ =
λ1 Eλ , . . . ,
λ∈Λ
λ d Eλ
(2.4)
λ∈Λ
Then S is a tuple of commuting operators which commutes with T, and T−S is nilpotent. For any multiindex α, we have Sα = S1α1 · · · Sdαd =
λα Eλ .
λ∈Λ
In the case V is a normed space and T is bounded, each operator in the tuple S is bounded as well. We now prove a very general result, which will provide affirmative answers to Questions 1 and 2 in the introduction.
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Theorem 2.6. Let X and Y be two d-tuples of commuting algebraic operators on a Hilbert space H. Let U (respectively, V) be the commuting tuple associated with X (respectively, Y) as in (2.4). Then Rad(J (A; X, Y)) ⊆ J (A; U, V).
(2.5)
Proof. Write X = [X1 , . . . , Xd ] and decompose H = ⊕λ∈Λ Hλ such that for each λ = (λ1 , . . . , λd ) ∈ Λ, the subspace Hλ is invariant for X and Xj − λj I is nilpotent on Hλ . Let Uλ denote the canonical projection from H onto Hλ . Then U = λ∈Λ λ · Uλ and for any multiindex α, we have Uα =
λα · Uλ .
λ∈Λ
Similarly, write Y = [Y1 , . . . , Yd ] and decompose H = ⊕ω∈Ω Kω . Let Vω be the canonical projection from H onto Kω . Then V = ω∈Ω ω · Vω and for any multiindex β, Vβ =
ω β · Vω .
ω∈Ω
Take any polynomial p ∈ Rad(J (A; X, Y)). For λ ∈ Λ, ω ∈ Ω and vectors u ∈ Hλ and v ∈ Kω , there exists an integer k ≥ 1 sufficiently large such that (Xj − λj I)k u = (Yj − ωj I)k v = 0 ¯ ω)Av, u = 0, which implies for all 1 ≤ j ≤ d. Proposition 2.4 shows that p(λ, ¯ ω)U ∗ AVω = 0. p(λ, λ Write p(x, y) =
α,β cα,β x
α β
y . We compute
p(A; U, V) =
cα,β U∗α AVβ
α,β
=
cα,β
α,β
=
λ∈Λ,ω∈Ω
=
λ∈Λ
β ¯αU ∗ A ω · V λ ω λ ω∈Ω
¯ α ω β U ∗ AVω cα,β λ λ
α,β
¯ ω)U ∗ AVω = 0. p(λ, λ
λ∈Λ,ω∈Ω
We conclude that p ∈ J (A; U, V). Since p ∈ Rad(J (A; X, Y)) was arbitrary, the proof of the theorem is complete. 2
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Theorem 2.6 enjoys numerous interesting applications that we now describe. Proof of Theorem 1.1. We shall prove the theorem under a more general assumption that T is a tuple of commuting algebraic operators. Since T is (A, m)-isometric, the polynod mial ( j=1 xj yj − 1)m belongs to the ideal J (A; T, T). It follows that the polynomial d p(x, y) = j=1 xj yj − 1 belongs to the radical ideal of J (A; T, T). By Theorem 2.6, we may decompose T = S + N, where N is a nilpotent tuple commuting with S and p(A; S, S) = 0, which means that S is a spherical A-isometry. 2 Example 2.7. Recall that an operator T is called (m, n)-isosymmetric (see [33]) if T is a hereditary root of f (x, y) = (xy − 1)m (x − y)n . Theorem 2.6 shows that any such algebraic T can be decomposed as T = S + N , where N is nilpotent, S is isosymmetric (i.e. (1, 1)-isosymmetric) and SN = N S. Example 2.8. Several researchers [27,9] have investigated the so-called toral m-isometric tuples. It is straightforward to generalize this notion to toral (A, m)-isometric tuples, which are commuting d-tuples T that satisfy |α| m1 , . . . , md (Tα )∗ A Tα = 0. (−1) α 0≤α1 ≤m1 ... 0≤αd ≤md
for all m1 +· · ·+md = m. Equivalently, T is a common hereditary root of all polynomials of the form (1 − x1 y1 )m1 · · · (1 − xd yd )md for m1 + · · · + md = m. This means that all these polynomials belong to the ideal J (A; T, T). We see that toral (A, 1)-isometries are just commuting tuples T such that each Tj is an A-isometry, that is, Tj∗ ATj = A. Note that for any toral (A, m)-isometry T, the radical ideal Rad(J (A; T, T)) contains all the polynomials 1 − xj yj : j = 1, 2, . . . , d . Theorem 2.6 asserts that T = S + N, where S is a toral (A, 1)-isometry and N is a nilpotent tuple commuting with S. 3. On 2-isometric tuples It is well known that any 2-isometry on a finite dimensional Hilbert space must actually be an isometry. On the other hand, there are many examples of finite dimensional 2-isometric tuples that are not spherical isometries. The following class of examples is given in Richter’s talk [31]. Example 3.1. If α = (α1 , . . . , αd ) ∈ ∂Bd and Vj : C m → C n such that then W = (W1 , . . . , Wd ) with Wj = defines a 2-isometric d-tuple.
αj In 0
Vj αj Im
d j=1
αj Vj = 0,
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The following result was stated in [31] without a proof and as far as the author is aware of, it has not appeared in a published paper. Theorem 3.2 (Richter-Sundberg). If T is a 2-isometric tuple on a finite dimensional space, then T = U ⊕ W, where U is a spherical unitary and W is a direct sum of operator tuples unitarily equivalent to those in Example 3.1. In this section, we shall assume that A is self-adjoint and investigate (A, 2)-isometric d-tuples. We obtain a characterization for such tuples that generalizes the above theorem. We first provide a generalization of Example 3.1. We call N = (N1 , . . . , Nd ) an (A, n)-nilpotent tuple if ANα = 0 for any indices α with |α| = n. Proposition 3.3. Assume that A is a self-adjoint operator. Let S be an (A, 1)-isometry and N an (A, 2)-nilpotent tuple such that S commutes with N. Suppose S1∗ AN1 + · · · + Sd∗ ANd = 0, then S + N is an (A, 2)-isometry. Proof. By the assumption, we have ANj Nk = Nj∗ Nk∗ A = 0 for 1 ≤ j, k ≤ d, d d d ∗ ∗ ∗ j=1 Sj ASj = A, and j=1 Sj ANj = j=1 Nj ASj = 0. It follows that d d (Sj + Nj )∗ A (Sj + Nj ) = A + Nj∗ ANj . j=1
j=1
We then compute
(Sk + Nk )∗ (Sj + Nj )∗ A (Sj + Nj )(Sk + Nk )
1≤k,j≤d
=
d
(Sk∗
+
Nk∗ )(A
+
d
Nj∗ ANj )(Sk + Nk )
j=1
k=1
=
d
(Sk∗ + Nk∗ )A (Sk + Nk ) +
k=1
=A+
(Sk∗ + Nk∗ )Nj∗ ANj (Sk + Nk )
1≤k,j≤d d
Nk∗ ANk +
k=1
=A+
d k=1
Sk∗ Nj∗ ANj Sk
1≤k,j≤d
Nk∗ ANk +
d j=1
Nj∗
d k=1
Sk∗ ASk Nj
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=A+
d
Nk∗ ANk +
d
Nj∗ ANj
j=1
k=1
=A+2
d
11
Nj∗ ANj
j=1
=2
d (Sj + Nj )∗ A (Sj + Nj ) − A. j=1
Consequently, the sum S + N is an (A, 2)-isometric tuple. 2 Remark 3.4. We have provided a direct proof of Proposition 2.3. Using the hereditary functional calculus and the approach in [26], one may generalize the result to the case S being an (A, m)-isometry and N an (A, n)-nilpotent commuting with S. Under such an assumption, if S1∗ AN1 + · · · + Sd∗ ANd = 0, then S + N is an (A, m + 2n − 3)-isometry. We leave the details for the interested reader. We now show that any algebraic (A, 2)-isometric tuple has the form given in Proposition 3.3 and as a result, provide a proof of Richter-Sundberg’s theorem. Theorem 3.5. Assume that A is a positive operator. Let T be an algebraic (A, 2)-isometric tuple on H. Then there exists an (A, 1)-isometric tuple S and a tuple N commuting with d S such that T = S + N, =1 S∗ AN = 0, and ANj N = 0 for all 1 ≤ j, ≤ d (we call such N an (A, 2)-nilpotent tuple). In the case H is finite dimensional and A = I, the identity operator, we recover Theorem 3.2. Proof. Recall that there exists a finite set Λ ⊂ C d and a direct sum decomposition H = ⊕λ∈Λ Hλ such that for each λ ∈ Λ, the subspace Hλ is invariant for T and Tj − λj I is nilpotent on Hλ . Let S be defined as in (2.4) and put N = T−S. From the construction, N is nilpotent and Theorem 2.6 shows that S is (A, 1)-isometric. We shall show that N satisfies the required properties. Restricting on each invariant subspace Hλ , we only need to consider the case H = Hλ and so S = λI. Proposition 2.4 asserts that (|λ|2 − 1)Av, u = 0 for all v, u ∈ H. If |λ| = 1, then A = 0 and the conclusion follows. Now we assume that |λ| = 1. Since N is nilpotent, there exists a positive integer r such that A Nα = 0 whenever |α| = r. We claim that r may be taken to be 2. To prove the claim, we assume r ≥ 3 and show that A Nα = 0 for all |α| = r − 1. Since T = λI + N is (A, 2)-isometric, the tuple N is an A-root of the polynomial
p(x, y) =
d d
2
2 ¯ j )(yj + λj ) − 1 = ¯ j yj . (xj + λ xj yj + λj xj + λ j=1
j=1
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On the other hand, N is an A-root of xα and yα for all |α| = r. This shows that p(x, y), xα and yα belong to J (A; N, N) for all |α| = r. To simplify the notation, we shall denote J (A; N, N) by J in the rest of the proof. Take any multiindex β with |β| = r − 2. We write xβ p(x, y)yβ = xβ
d
λj x j
d
j=1
¯ y yβ + xγ Hγ (x, y) + Gγ (x, y) yγ λ |γ|≥r
=1
for some polynomials Hγ and Gγ . Since the left-hand side and the second term on the right-hand side belong to J , which is an ideal, we conclude that xβ
d
λj x j
d
j=1
¯ y yβ ∈ J . λ
=1
d ¯ y yβ and xβ d λj xj are in J . Now Proposition 2.3 shows that both λ =1 j=1 for any multiindex γ with |γ| = r − 3, we compute xγ p(x, y)yγ = xγ
d
d
2 xj yj yγ + xβ λj xj Pβ (x, y) |β|=r−2
j=1
+
d |β|=r−2
j=1
λj xj yβ Qβ (x, y).
j=1
Since the left-hand side and the last two sums on the right-hand side belong to J , it d follows that xγ ( j=1 xj yj )2 yγ belongs to J . Another application of Proposition 2.3 then shows that yj y yγ belongs to J for all 1 ≤ j, ≤ d. That is, yα belongs to J whenever |α| = r − 1 (as long as r ≥ 3). As a consequence, we see that yα , and hence xα , belong to J for all |α| = 2. This together with the fact that p(x, y) ∈ J forces d d ¯ d ¯ ( j=1 λj xj )( =1 λ y ) to belong to J , which implies that =1 λ y is in J . We have d ¯ d ∗ then shown ANj N = 0 for all 1 ≤ j, ≤ d and S AN = λ AN = 0, as =1
=1
desired. Now let us consider T a 2-isometric tuple on a finite dimensional space H. Recall that we have the decomposition H = ⊕λ∈Λ Hλ such that for each λ ∈ Λ, the subspace Hλ is invariant for T and Tj − λj I is nilpotent on Hλ . By Proposition 2.4, we have (ω, λ − 1)2 v, u = 0 for all v ∈ Hω and u ∈ Hλ . It follows that |λ| = 1 for all λ ∈ Λ and Hλ ⊥ Hω whenever λ = ω. As a result, each subspace Hλ is reducing for T. To complete the proof, it suffices to consider H = Hλ . We shall show that either T is a spherical unitary or it is unitarily equivalent to a tuple given in Example 3.1. Indeed, d ¯ we have T = λI + N, where =1 λ N = 0 and Nj N = 0 for all 1 ≤ j, ≤ d. If N = 0, then T is a spherical unitary. Otherwise, let M = ker(N1 ) ∩ · · · ∩ ker(Nd ). Then N (H) ⊆ M for all 1 ≤ ≤ d. As a consequence, with respect to the orthogonal decomposition H = M ⊕ M⊥ , each N has the form
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N =
0 V 0 0
13
d ¯ d ¯ for some V : M⊥ → M. Since =1 λ N = 0, we have =1 λ V = 0. It follows that T is unitarily equivalent to an operator tuple in Example 3.1. 2 Acknowledgments The author wishes to thank the referee for helpful suggestions which improved the presentation of the paper. References [1] J. Agler, Subjordan operators, PhD Thesis, ProQuest LLC, Indiana University, Ann Arbor, MI, 1980. [2] J. Agler, Sub-Jordan operators: Bishop’s theorem, spectral inclusion, and spectral sets, J. Oper. Theory 7 (2) (1982) 373–395. [3] J. Agler, A disconjugacy theorem for Toeplitz operators, Am. J. Math. 112 (1) (1990) 1–14. [4] J. Agler, J.W. Helton, M. Stankus, Classification of hereditary matrices, Linear Algebra Appl. 274 (1998) 125–160. [5] J. Agler, M. Stankus, m-isometric transformations of Hilbert space I, Integral Equ. Oper. Theory 21 (4) (1995) 383–429. [6] J. Agler, M. Stankus, m-isometric transformations of Hilbert space II, Integral Equ. Oper. Theory 23 (1) (1995) 1–48. [7] J. Agler, M. Stankus, m-isometric transformations of Hilbert space III, Integral Equ. Oper. Theory 24 (4) (1996) 379–421. [8] O.A.M. Sid Ahmed, A. Saddi, A-m-isometric operators in semi-Hilbertian spaces, Linear Algebra Appl. 436 (10) (2012) 3930–3942. [9] A. Athavale, V. Sholapurkar, Completely hyperexpansive operator tuples, Positivity 3 (3) (1999) 245–257. [10] T. Bermúdez, A. Saddi, H. Zaway, (A, m)-isometries on Hilbert spaces, Linear Algebra Appl. 540 (2018) 95–111. [11] T. Bermúdez, A. Martinón, V. Müller, J.A. Noda, Perturbation of m-Isometries by nilpotent operators, Abstr. Appl. Anal. (2014) 745479. [12] T. Bermúdez, A. Martinón, J.A. Noda, An isometry plus a nilpotent operator is an m-isometry. Applications, J. Math. Anal. Appl. 407 (2) (2013) 505–512. [13] T. Bermúdez, A. Martinón, J.A. Noda, Products of m-isometries, Linear Algebra Appl. 438 (1) (2013) 80–86. [14] F. Botelho, On the existence of n-isometries on p spaces, Acta Sci. Math. (Szeged) 76 (1–2) (2010) 183–192. [15] M. Ch¯ o, S. Ôta, K. Tanahashi, Invertible weighted shift operators which are m-isometries, Proc. Am. Math. Soc. 141 (12) (2013) 4241–4247. [16] B.P. Duggal, Tensor product of n-isometries, Linear Algebra Appl. 437 (1) (2012) 307–318. [17] B.P. Duggal, Tensor product of n-isometries III, Funct. Anal. Approx. Comput. 4 (2) (2012) 61–67. [18] M. Faghih-Ahmadi, K. Hedayatian, m-isometric weighted shifts and reflexivity of some operators, Rocky Mt. J. Math. 43 (1) (2013) 123–133. [19] M. Faghih-Ahmadi, Powers of A-m-isometric operators and their supercyclicity, Bull. Malays. Math. Sci. Soc. 39 (3) (2016) 901–911. [20] J. Gleason, S. Richter, m-isometric commuting tuples of operators on a Hilbert space, Integral Equ. Oper. Theory 56 (2) (2006) 181–196. [21] C. Gu, Examples of m-isometric tuples of operators on a Hilbert space, J. Korean Math. Soc. 55 (1) (2018) 225–251. [22] C. Gu, M. Stankus, Some results on higher order isometries and symmetries: products and sums with a nilpotent operator, Linear Algebra Appl. 469 (2015) 500–509.
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