Approximation of the polar factor of an operator acting on a Hilbert space

Journal Pre-proof Approximation of the polar factor of an operator acting on a Hilbert space

Mostafa Mbekhta

PII:

S0022-247X(20)30116-5

DOI:

https://doi.org/10.1016/j.jmaa.2020.123954

Reference:

YJMAA 123954

To appear in:

Journal of Mathematical Analysis and Applications

Received date:

26 September 2019

Please cite this article as: M. Mbekhta, Approximation of the polar factor of an operator acting on a Hilbert space, J. Math. Anal. Appl. (2020), 123954, doi: https://doi.org/10.1016/j.jmaa.2020.123954.

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Approximation of the polar factor of an operator acting on a Hilbert space Mostafa Mbekhta Universit´ e de Lille, D´ epartement de Math´ ematiques, UMR-CNRS 8524, Labo. P. Painlev´ e 59655 Villeneuve d’Ascq Cedex France.

Abstract Let H be a complex Hilbert space and let B(H) be the algebra of all bounded linear operators on H. The polar decomposition theorem asserts that every operator T ∈ B(H) can be written as the product T = V P of a partial isometry V ∈ B(H) and a positive operator P ∈ B(H) such that the kernels of V and P coincide. Then this 1 decomposition is unique. V is called the polar factor of T . Moreover, we have automatically P = |T | = (T ∗ T ) 2 . Unlike P , we do not have any representation formula for V . In this paper, we give several explicit formulas representing the polar factor. These formulas allow for methods of approximations of the polar factor of T . Keywords: 2010 MSC: 47B48, 47A10, 46H05 polar decomposition, polar factor, partial isometries, approximations. This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01)

1. Introduction Throughout this paper, let H be a complex Hilbert space and let B(H) be the algebra of all bounded linear operators on H. For an arbitrary operator T ∈ B(H), we denote by σ(T ), R(T ), N (T ) and T ∗ the spectrum, the range, the null subspace and the adjoint operator of T , respectively. For any closed subspace M of H, let PM denote the orthogonal projection onto M . An operator T ∈ B(H) is a partial isometry when T T ∗ T = T (or, equivalently, T ∗ T is an orthogonal projection; in this case, T ∗ T = PN (T )⊥ ). In particular, T is an isometry if T ∗ T = I, and T is unitary if it is a surjective isometry. The well known polar decomposition of any operator T ∈ B(H) is a generalization of the polar decomposition of a nonzero complex number z = exp(iθ) |z|, θ ∈ R, and consists of writing T as a product T = V P,

(1)

of a partial isometry V ∈ B(H) and a positive operator P ∈ B(H) such that N (V ) = N (P ). Then P = |T | := 1 (T ∗ T ) 2 is the modulus of T and N (V ) = N (T ). Therefore the decomposition (1.1) becomes T = V |T | and it is unique. The operator V is called the polar factor of T . In this paper we are interested in the approximation of the polar factor of an operator acting on a Hilbert space. The results obtained are motivated by the work of N.J.Higham. Indeed, in [8, 9], N.J.Higham shows the following: for A a nonsingular matrix, by considering the iteration sequence defined by X0 = A, Xk+1 =

1 (Xk + Xk∗−1 ), 2

that each iterate Xk is nonsingular and limk→∞ Xk = U , where U is the unitary polar factor in the polar decomposition of A. On the other hand, one can show the existence and uniqueness of the polar decomposition Email address: [email protected] (Mostafa Mbekhta)

Preprint submitted to Elsevier

February 12, 2020

by approximation methods in the general context of von Neumann algebras; see, e.g., [12, Proposition 2.2.9] or [14, Theorem 1.12.1]. The paper is organized as follows. In Section 2 we give an approximation of the polar part by an integral. We complete the result [4, Theorem 2.1] by giving more details on the convergence, in the case of operators with closed range. We obtain some results on the approximation of the polar factor by Newton interpolation polynomials in Section 3, and by Hermite interpolation polynomials in Section 4. The last section is devoted to a discussion on the approximation of the polar factor of an operator by some partial isometries. 2. Approximation of the polar factor by an integral We motivate the idea of our approach to state Theorem 2.1. Let us observe that if T = V |T | is the polar decomposition of T , then we have T ∗ V − |T | = 0. This equality suggests the problem of minimizing the following function on H: F (x) = T ∗ x − |T |y2 =< T ∗ x − |T |y, T ∗ x − |T |y >, where y ∈ H fixed. Clearly, for y ∈ H fixed, the minimum of F (x) is attained at x = V y. Now, consider x as a function x(t), t ≥ 0, with x(0) = 0. By differentiating the function F (x(t)), we obtain d F (x(t)) dt

= =

d x(t) > dt d 2Re < T (T ∗ x(t) − |T |y), x(t) > . dt

2Re < T ∗ x(t) − |T |y, T ∗

If we take d x(t) dt

=

−T (T ∗ x(t) − |T |y) = −T T ∗ x(t) + T |T |y,

(2)

we get that d F (x(t)) = −2T (T ∗ x(t) − |T |y) 2 < 0. dt It follows that F (x(t)) is a decreasing function in t, asymptotically approaching its infimum when t tends to infinity. Then x(t) approaches V y. Therefore by the resolution of the differential equation (2.1), we obtain  t  t ∗ x(t) = exp((t − s)T T )T |T |y ds = exp(−sT T ∗ )T |T |y ds. 0

0

Taking t → +∞, we get, for arbitrary y ∈ H, Vy =

 0

∞

exp(−sT T ∗ )T |T |y ds.

This motivates the following theorem which gives a new proof of the existence and uniqueness of the polar decomposition of a bounded operator on a complex Hilbert space. Theorem 2.1. ([4, Theorem 2.1]) For each T ∈ B(H), there is a unique partial isometry V ∈ B(H) such that T = V |T | and V ∗ V = PN (T )⊥ . Furthermore, we have  ∞  ∞ exp(−sT T ∗ )T |T | ds = T exp(−sT ∗ T )|T | ds, V = 0

0

with convergence in the strong operator topology of B(H).

2

In the case where T has closed range, we obtain an additional result which gives more precision on the convergence. To do so, we need to introduce the reduced minimum modulus that measures the closedness of the range of an operator. Recall that the reduced minimum modulus of an operator T ∈ B(H) is defined by ⎧   if T = 0 ⎨ inf T x; x = 1, x ∈ N (T )⊥ γ(T ) := ⎩ +∞ if T = 0. The reduced minimum modulus of an operator T ∈ B(H) has the following properties. i. T has closed range if and only if γ(T ) > 0. ii. γ(T ) = inf σ(|T |) \ {0}; iii. The following equalities are valid : γ(T )2 = γ(T ∗ T ) = γ(T T ∗ ) = γ(T ∗ )2 . iv. In particular, if T has closed range, then σ(T ∗ T ) ∪ σ(T T ∗ ) ⊆ {0} ∪ [γ(T )2 , T 2 ]. For further information, the interested reader may consult [3, 10]. The following theorem gives more precision on the convergence in Theorem 2.1, when the range of T is closed. Theorem 2.2. If T ∈ B(H) has closed range, then there is a unique partial isometry V ∈ B(H) such that T = V |T |, V ∗ V = PN (T )⊥ and which satisfies A(t) − V  ≤ exp(−tγ(T )2 ), t where A(t) = 0 T exp(−sT ∗ T )|T | ds. Therefore, the polar factor of the polar decomposition of T is given by  ∞ V = T exp(−sT ∗ T )|T | ds, 0

with convergence in the norm topology of B(H). Proof. If t > t, then we have (A(t ) − A(t))∗ (A(t ) − A(t))



t

= 

t



t

t

=

exp(−sT ∗ T )|T |T ∗ ds

 t

∗

∗

∗

∗

exp(−sT T ) ds (T T ) 

t

2

∗

∗

exp(−sT T )T T ds 2

(exp(−tT ∗ T ) − exp(−t T ∗ T )) .

3



t

∗

exp(−sT T ) ds t t

∗

exp(−sT T ) ds t

t

=

T exp(−sT ∗ T )|T | ds





2

t

=



exp(−sT T ) ds |T |T T |T |

t

=

t

It follows that A(t ) − A(t)

=  exp(−tT ∗ T ) − exp(−t T ∗ T ) =  exp(−tT ∗ T )(I − exp(−(t − t)T ∗ T )) = sup {exp(−tλ)(1 − exp(−(t − t)λ))}. λ∈σ(T ∗ T )

Since T has closed range, σ(T ∗ T ) ⊆ {0} ∪ [γ(T )2 , T 2 ]. Hence A(t ) − A(t) ≤ ≤ =

sup λ∈[γ(T )2 ,T 2 ]

sup λ∈[γ(T )2 ,T 2 ]

{exp(−tλ)(1 − exp(−(t − t)λ))} (since 0 < exp(−(t − t)λ) ≤ 1)

{exp(−tλ)}

exp(−tγ(T )2 ).

Hence, A(t ) − A(t) ≤ exp(−tγ(T )2 ).

(3)

Therefore, the net A(t) converges in the norm topology to an element V in B(H). On the other hand, 2

A(t)|T | − T 

=

=

  t

2   ∗ ∗ T  exp(−sT T )T T ds − I   0  ⎛ ⎞2   n+1    t n ∗ n+1 T ⎝ ⎠ −I  (−1) (T T )   (n + 1)!   n≥0 2

= T exp(−tT ∗ T ) = exp (−2tT ∗ T ) T ∗ T  = sup λ exp(−2tλ). λ∈σ(T ∗ T )

Hence, letting t → +∞, we obtain T = V |T |. Let P = PN (T )⊥ be the orthogonal projection onto N (T )⊥ . Then P = PR(|T |) which implies that P |T | = |T | = |T |P . Therefore A(t)P = A(t) and hence, V P = V . We have |T |P |T | = T ∗ T = |T |V ∗ V |T |. It follows that |T |(P − V ∗ V )|T | = 0. Since N (|T |) = N (P ), we get P (P − V ∗ V )P = 0 and P = P V ∗ V P . On the other hand, since V P = V , we have V ∗ V P = V ∗ V = P V ∗ V . Thus, P = P V ∗ V P = P V ∗ V = V ∗ V P = V ∗ V . Therefore, V ∗ V = P , and hence V is a partial isometry such that N (V ) = N (P ) = N (T ). Finally, to prove the uniqueness, suppose that T = V |T | = V  |T | are polar decompositions of T . Then  (V − V )|T | = 0. Hence, V  = V on R(|T |) and since N (V  ) = N (|T |) = N (V ), we conclude that V  = V . Now, when t tends to infinity in (2.2), we get A(t) − V  ≤ exp(−tγ(T )2 )

for all t ≥ 0. 

The proof is complete. 3. Approximation of the polar factor by Newton interpolation polynomials In this section we shall describe the polar factor by means of Newton interpolation polynomials.

4

First we shall consider the divided differences interpolation polynomial of f (x) = x1 . For i = 1, 2, ..., n, let n   k k k−1 Pn (i) = k=0 i−1 f (i)). In particular, for f (i) = 1i k Δ f (1) where Δf (i) := f (i + 1) − f (i), Δ f (i) := Δ(Δ it is known that Δk f (1) =

(−1)k k+1 .

Then, in this case, we obtain Pn (x) =

n  k=0

where, by convention,

−1

j=0 (1

−

x j+1 )

k−1 x 1  ) (1 − k + 1 j=0 j+1

= 1, so P0 (x) = 1 (see [6, 13]).

Theorem 3.1. For each T ∈ B(H), there is a unique partial isometry V ∈ B(H) such that T = V |T | and V ∗ V = PN (T )⊥ . Furthermore, we have V =

∞  k=0

k−1 ∞ k−1  T ∗T TT∗ T  1  )|T | = )|T ∗ |T, (I − (I − k + 1 j=0 j+1 k + 1 j=0 j+1

(4)

k=0

where the convergence is in the strong topology of B(H). Before we prove Theorem 3.1, we present some technical results. Lemma 3.1. For all λ ≥ 0 and integers n ≥ 0, λPn (λ) = 1 −

n 

(1 −

j=0

λ ). j+1

Proof. For λ = 0 the result is obvious. Suppose that λ > 0. We proceed by induction. For n = 0, the result is clear. Suppose that it is true for n. Then we have λPn+1 (λ)

=

=

=

=

λPn (λ) +

n λ  λ λ )+ ) (1 − j + 1 n + 2 j + 1 j=0 j=0   n λ λ 1− 1− ) (1 − n + 2 j=0 j+1

1−

1−

n 

n λ  λ ) (1 − n + 2 j=0 j+1

(1 −

n+1 

(1 −

j=0

λ ) j+1

Therefore, for all λ ≥ 0 and integers n ≥ 0, we have λPn (λ) = 1 − Lemma 3.2. For every m ∈ N,

⎧ 0 λ m ⎨ lim ) = (1 − n→∞ ⎩ j+1 j=0 1 n 

n

j=0 (1

−

λ j+1 ).



if λ > 0 if λ = 0.

Proof. For λ = 0 the result is obvious. Suppose that λ > 0. For every λ ∈ R+ we have 1 − λ ≤ e−λ , so 1 λ ≤ e−λ j+1 for every λ ∈ R+ and j ∈ N. Hence, there exists J0 ∈ N such that for every j ≥ J0 we have 1 − j+1 0≤1−

1 λ ≤ e−λ j+1 . j+1

5

From this we obtain 0≤ Hence, by setting tn :=

(1 −

j=J0

n

1 j=J0 j+1

0≤

n 

n λ ) ≤ e−λ j=J0 j+1

1 j+1

.

we obtain

n 

(1 −

j=J0

λ m ) ≤ e−mλtn j+1

Since λ > 0, m > 0 and limn→+∞ tn = +∞, λ m j+1 ) = 0.

limn→∞

for every n, m ∈ N.

n

j=J0 (1

−

λ m j+1 )

(5)

= 0. Therefore, limn→∞

n

j=0 (1

− 

Lemma 3.3. For any operator T ∈ B(H) and any f continuous function f on the compact set σ(T ∗ T ) ∪ {0} = σ(T T ∗ ) ∪ {0}, we have T f (T ∗ T ) = f (T T ∗ )T. In particular, T |T | = |T ∗ |T,

(6)

and for all n ≥ 0 T Pn (T ∗ T ) = Pn (T T ∗ )T

T Pn (T ∗ T )T ∗ = Pn (T T ∗ )T T ∗ .

and thus

(7)

Proof. Observe that T (T ∗ T ) = (T T ∗ )T . It follows that for any polynomial P ∈ C[X], we have T P (T ∗ T ) = P (T T ∗ )T . Using the Stone-Weierstrass theorem, we deduce that T f (T ∗ T ) = f (T T ∗ )T. 

So the first equality is shown. The other equalities follow easily. Proof. (Theorem 3.1) Put Vn = T Pn (T ∗ T )|T |, n ≥ 0. Then for m > n, we have Xn,m := (Vm − Vn )(Vm − Vn )∗

=

T (Pm (T ∗ T ) − Pn (T ∗ T )) T ∗ T (Pm (T ∗ T ) − Pn (T ∗ T )) T ∗

= =

[T (Pm (T ∗ T ) − Pn (T ∗ T )) T ∗ ]2 [(Pm (T T ∗ ) − Pn (T T ∗ )) T T ∗ ]2 (Lemma 3.3) ⎡ ⎤2 m n ∗   TT∗ ⎦ ⎣ (I − T T ) − ) (I − (Lemma 3.1) j+1 j+1 j=0 j=0

=

⎡ =

⎣

n  j=0

(I −

∗

m 

∗

⎤2

TT TT )(I − ))⎦ . (I − j+1 j +1 j=n+1

Now, by spectral theory and Lemma 3.2, we see that Xn,m tends weakly to zero. Since the sequence Xn,m is positive, it converges strongly to zero (see [12, page 20, line 9]). It follows that Vn is strongly convergent to an element V in B(H). On the other hand, using Lemma 3.3, we obtain Vn |T | − T = T Pn (T ∗ T )T ∗ T − T = (T Pn (T ∗ T )T ∗ − I)T = (Pn (T T ∗ )T T ∗ − I)T.

6

It follows that Vn |T | − T 2

(Pn (T T ∗ )T T ∗ − I)T T ∗ (Pn (T T ∗ )T T ∗ − I) (Pn (T T ∗ )T T ∗ − I)2 T T ∗  ⎛ ⎞2 n ∗  TT ⎠ =  ⎝ (I − ) T T ∗ ( by Lemma 3.1) j + 1 j=0

= =

=

sup λ∈σ(T T ∗ )

≤

sup λ∈σ(T T ∗ )

λ

n 

(1 −

j=0

λ 2 ) j+1

λ exp(−2λtn )

(by (3.2)).

Hence, letting n → +∞, we obtain T = V |T |. Let us show that V is a partial isometry with N (V ) = N (T ). Let P = PN (T )⊥ be the orthogonal projection onto N (T )⊥ . Then |T |P = |T | = P |T |. Therefore, Vn P = Vn , and hence V P = V . It follows that |T |P |T | = T ∗ T = |T |V ∗ V |T | and thus |T |(P − V ∗ V )|T | = 0. Since N (|T |) = N (P ), we get P (P − V ∗ V )P = 0 and P = P V ∗ V P . Clearly, V P = V implies V ∗ V P = V ∗ V = P V ∗ V . Thus, P = P V ∗ V P = P V ∗ V = V ∗ V P = V ∗ V . Therefore, V ∗ V = P , and hence V is a partial isometry such that T = V |T | and N (V ) = N (P ) = N (T ). The uniqueness of V follows as in the proof of Theorem 2.2.  Remark 3.1. Clearly, Theorem 3.1 remains true in the more general context of von Neumann algebras. The polynomials Pn (x) can be rewritten in the following way: P0 (x) = 1, Pn+1 (x) = Pn (x) +

1 [1 − xPn (x)], n ≥ 1. n+2

In fact, by Lemma 2.1, we have Pn+1 (x) = Pn (x) +

n 1  x 1 ) = Pn (x) + [1 − xPn (x)]. (1 − n + 2 j=0 j+1 n+2

Hence, given T ∈ B(H), let us define V0 = T |T |, Vn+1 := Vn +

1 (V0 − T T ∗ Vn ), n ≥ 1. n+2

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Thus, as an immediate consequence of Theorem 3.1, we obtain the following corollary: Corollary 3.1. If T ∈ B(H), then the sequence (Vn )n≥0 defined by (3.5) converges to V ∈ B(H), the polar factor of T , in the strong topology of B(H). The following theorem gives more precision on the convergence in Theorem 3.1, when the range of T is closed. Theorem 3.2. If T ∈ B(H) has closed range, then the convergences in Theorem 3.1 and Corollary 3.1 are in the uniform topology of B(H). Furthermore, there is a constant C > 0, independent of n, such that Vn − V  ≤ C

1 (for n sufficiently large). (n + 2)γ(T )2

7

Proof. Assume that T has closed range. Then γ(T ) > 0 and σ(T T ∗ ) ⊆ {0} ∪ [γ(T )2 , T 2 ]. From the proof of Theorem 3.1, we have Vm − Vn 

= 

n 

(I −

j=0

=

m  TT∗ TT∗ )(I − )) (I − j+1 j+1 j=n+1

|

sup

n 

λ∈σ(T T ∗ ) j=0

≤

(1 − |

sup

n 

λ∈[γ(T )2 ,T 2 ] j=0

m  λ λ )(1 − ))| (1 − j+1 j + 1 j=n+1

(1 −

m  λ λ )(1 − ))|. (1 − j+1 j + 1 j=n+1

Consequently, the sequence (Vn )n converges to V in the uniform topology of B(H). 2 , then for all j ≥ J, λ ∈ [γ(T )2 , T 2 ], Note that, if J +1 = E(T 2 ), where E(T 2 ) is the integer part of T m λ λ ) < 1. Hence 0 < 1 − j+1 < 1. Therefore, for n, m sufficiently large, we have 0 < 1 − j=n+1 (1 − j+1 Vm − Vn  ≤

|

sup

n 

λ∈[γ(T )2 ,T 2 ] j=0

(1 −

λ ). j+1

It follows that Vn − V  ≤

Now,

n

λ j=0 j+1 . n 1 since j=J j+1

where tn =

≥

 n+2 J+1

dt t

exp(−λ

sup λ∈[γ(T )2 ,T 2 ]

exp(−λtn ) (for n sufficiently large),

= ln(n + 2) − ln(J + 1), we get

n  j=J

λ ) ≤ (J + 1)λ (n + 2)−λ ≤ T 2λ (n + 2)−λ . j+1

Therefore, if we set C = maxλ∈σ(T T ∗ ) T 2λ | Vn − V  ≤ C

J

j=0 (1

−

λ j+1 )|,

then

1 (for n sufficiently large). (n + 2)γ(T )2 

The proof is complete. 4. Approximation of the polar factor by Hermite interpolation polynomials

In a similar manner, the polar factor can also be described by means of the Hermite interpolation polynomial of f (x) = x1 . This interpolation polynomial is given by Qn (x) =

n 

[2(1 + k) − x]

k=0

k−1  1 x 2 ) for n ≥ 0, (1 − 2 (1 + k) j=0 j+1

−1 x 2 ) = 1, so Q0 (x) = 2 − x. where, by convention, j=0 (1 − j+1 The Hermite interpolation polynomial is the unique polynomial of degree 2n+1 such that Qn (xi ) = f (xi ) = and Qn (xi ) = f  (xi ) = −1 for xi = i + 1 with i = 0, 1, 2, ..., n (see [6, 13]). x2 i

Lemma 4.1. For all λ ≥ 0 and integers n ≥ 0, λQn (λ) = 1 −

n  j=0

8

(1 −

λ 2 ) . j+1

1 xi

Lemma 4.2. Let T ∈ B(H). Then, for all integers n ≥ 0, T Qn (T ∗ T )T ∗ = Qn (T T ∗ )T T ∗ . The proof of the following theorem is obtained in the same way as that of Theorem 3.1. Theorem 4.1. For each T ∈ B(H), there is a unique partial isometry V ∈ B(H) such that T = V |T | and V ∗ V = PN (T )⊥ . Furthermore, we have V =

∞ 

[2(1 + k)I − T T ∗ ]

k=0

k−1  T T ∗T 2 ) |T |, (I − 2 (1 + k) j=0 1+j

(9)

with convergence in the strong topology of B(H). The Hermite polynomials can also be defined as Q0 (x) = 2 − x, Qn+1 (x) = Qn (x) +

x 1 (2 − )[1 − xQn (x)], n ≥ 1. n+2 n+2

In fact, Qn+1 (x)

=

n  1 x 2 ) Qn (x) + [2(n + 2) − x] (1 − (n + 2)2 1 + j j=0

= Qn (x) +

x 1 [2 − ][1 − xQn (x)]. (n + 2) n+2

Hence, given T ∈ B(H), let us define V0 = (2 − T T ∗ )T |T |, Vn+1 = Vn +

TT∗ 1 (2I − )[T |T | − T T ∗ Vn ], n ≥ 1. n+2 n+2

(10)

Thus, as an immediate consequence of Theorem 4.1, we obtain the following corollary: Corollary 4.1. If T ∈ B(H), then the sequence (Vn )n≥0 defined by (4.2) converges to V ∈ B(H), the polar factor of T , in the strong topology of B(H). In the case where T has closed range, we obtain the following result which gives more precision on the convergence. The proof is obtained in the same way as that of Theorem 3.2 Theorem 4.2. If T ∈ B(H) has closed range, then the convergence in Theorem 4.1 and Corollary 4.1 are in the uniform topology of B(H). Furthermore, there is a constant C > 0, independent of n, such that Vn − V  ≤ C

1 (for n sufficiently large). (n + 2)2γ(T )2

9

5. Approximation of the polar factor by partial isometries We conclude this article by a discussion concerning the approximation of the polar factor by partial isometries. Several authors have investigated the problem of approximation by partial isometries. The interested reader can consult [1, 2, 5, 7, 8, 9, 11, 15] and the references therein. For T ∈ B(H), denote ΛT = {W ∈ B(H); W is a partial isometry with N (W ) = N (T )}. Then we have the following conjecture that would present the polar factor V of T as the minimum of the function F : ΛT → [0, +∞[ defined by F (W ) = T − W ,

W ∈ ΛT .

Conjecture 5.1. If T ∈ B(H) and V ∈ ΛT , then the following conditions are equivalent: (i) V is the polar factor of T ; (ii) T − V  = min{T − W ; W ∈ ΛT }. When T is injective then ΛT becomes the set of all isometries. The proof of the following result is largely inspired by the arguments developed in the proof of [1, Theorem 3.1]. Theorem 5.1. If T ∈ B(H) is injective and T = V |T | the polar decomposition of T , then T − V  = min{T − W ; W ∈ ΛT }. Proof. Clearly we have T − V  ≥ min{T − W ; W ∈ ΛT }.

(11)

Let us show the other inequality. Since V is an isometry, we have : T − V  = V (|T | − I) =  I − |T |.

(12)

To complete the proof we need the following well-known lemma. Lemma 5.1. Let S ∈ B(H) be a positive operator. Then I − S = sup | 1 − Sx |. x=1

Proof. The proof is a direct consequence of the fact that inf Sx = inf < Sx, x >

x=1

x=1

and

sup Sx = sup < Sx, x > .

x=1

x=1

 Now, let W ∈ ΛT and x ∈ H with x = 1. Then (W − T )x

≥ | W x − T x | = | 1 −  |T |x |.

Therefore, using Lemma 5.1, for every W ∈ ΛT we obtain (W − T )

≥

 I − |T | .

Finally, from (5.1), (5.2) and (5.3), we deduce the desired result.

10

(13) 

As an immediate consequence of Theorem 5.1, we obtain the following corollary: Corollary 5.1. If T ∈ B(H) is surjective and T = V |T | the polar decomposition of T , then T − V  = min{T − W ; W ∗ ∈ ΛT ∗ }. Proof. The proof follows by applying the above theorem to T ∗ .



Remark 5.1. (1) Theorem 5.1 and Corollary 5.1 give a partial answer to Conjecture 5.1. (2) Note that in finite dimension, the authors of [11] show the following result : Theorem 5.2. ([11]). Let T ∈ Cm×n have the polar decomposition T = V |T |. Then T − V  = min{T − W ; W is a partial isometry with R(W ) = R(T )}. References References [1] J.G. Aiken, J.A. Erdos and J.A. Goldstein, Unitary approximation of positive operators, Illinois J. Math., vol. 24 (1980), 61-72. [2] J. Antezana, E. Chiumiento, Approximation by partial isometries and symmetric approximation of finite frames, J. Fourier Anal. Appl. 24 (2018), no. 4, 1098-1118. [3] C. Apostol, The reduced minimum modulus, Michigan Math. J. 32 (1985), 279-294. [4] F. Chabbabi, M.Mbekhta, Polar decomposition, Aluthge and mean transforms, 2019 (to apear). [5] E. Chiumiento, Global symmetric approximation of frames, J. Fourier Anal. Appl. (2019), in press. [6] J.P. Demailly, Analyse Num´erique et Equations Diff´erentielles, Presses universitaires de Grenoble, Grenoble, 1991. [7] M. Frank, V. Paulsen, T. Tiballi, Symmetric approximation of frames and bases in Hilbert spaces, Trans. Am. Math. Soc. 354, (2002), 777-793. [8] N.J. Higham, Computing the polar decomposition with applications, SIAM. J. Stat. Comput. Vol. 7 (1986), 1160-1174. [9] N.J. Higham, Functions of Matrices Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA (2008). [10] T. Kato, Perturbation Theory for Linear Operators , Springer, Berlin 1980. [11] Laszkiewicz, B.; Zietak, K. Approximation of matrices and a family of Gander methods for polar decomposition. BIT vol. 46 (2006), 345-366. [12] G. K. Pedersen, C ∗ -Algebras and their Automorphism Groups, Academic Press INC. (London) 1979. [13] A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics 2nd Edition, Springer, Berlin, 2007. [14] S. Sakai, C ∗ -algebras and W ∗ -algebras, Springer Verlag. Berlin 1971. [15] P.Y. Wu, Approximation by partial isometries, Proc. Edinb. Math. Soc. 29, (1986), 255-261.

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