A Spectral Approach to Distributional Extensions of Normal Operators

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

200, 224]244 Ž1996.

0200

A Spectral Approach to Distributional Extensions of Normal Operators F. Baptiste, W. Lamb, and D. F. McGhee Department of Mathematics, Uni¨ ersity of Strathclyde, Glasgow, G1 1XH, Scotland, United Kingdom Submitted by H. M. Sri¨ asta¨ a Received January 23, 1995

A constructive method for producing a test function space and hence a generalized function space suitable for a given normal operator is developed. Spectral theory and a functional calculus are shown to be valid in these spaces. Application of the theory is illustrated by considering a normal realisation of the operator x Ž drdx . and demonstrating that the test and generalized function spaces for this operator coincide with certain spaces originally introduced by McBride in the theory of fractional calculus. It is shown that the so-called Hadamard operators arise as functions of x Ž drdx . and so a distributional theory of these operators can be obtained using the functional calculus in the generalized function space. Q 1996 Academic Press, Inc.

1. INTRODUCTION Recently, Lamb and McGhee w6x extended the work of Zemanian w13x and Judge w5x to develop a spectral theory for the extension, to an appropriate space of distributions, of an arbitrary self-adjoint operator in a separable Hilbert space. A functional calculus was shown to be valid in this setting and it was applied, via semigroup theory, to the solution of distributional initial-boundary value problems. Here, we consider a similar approach for a normal operator in a separable Hilbert space. A preliminary report of such a theory, developed within the topological framework of a countably Hilbert space, appears in w10x. However, in this paper, we revert to the multinormed approach used in w6x since this is a more convenient form for demonstrating links with distributional results obtained by other authors. 224 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

EXTENSIONS OF NORMAL OPERATORS

225

A motivating factor for our work is that it provides a constructi¨ e method for producing test function spaces, and hence generalized function spaces, tailored to particular normal operators. As we shall demonstrate, the spaces produced for a specific normal operator T prove to be equally suitable for a large class of functions of T defined via spectral integrals. This fact can be exploited by using a relatively simple normal operator T to produce, systematically, a space of generalized functions ideally suited to more complicated normal operators F ŽT .. To illustrate these ideas, we examine a particular differential operator and demonstrate that the spaces constructed coincide with certain test and generalized function spaces introduced by McBride w8x in work on Mellin multipliers and fractional calculus. By considering fractional powers of this normal differential operator, we also show how a distributional theory for logarithmic fractional integrals, or Hadamard operators w4, p. 672x, can be obtained directly from our theory.

2. SPACES OF TEST AND GENERALIZED FUNCTIONS Let H be a separable complex Hilbert space with inner product Ž?,? ., linear in its first argument, and norm 5 ? 5 [ Ž ?,?. . Given a normal operator T : DŽT . ; H ª H, we consider the set

'

D[

F DŽ T k . ; H

kgN 0

where N0 [  0, 1, 2, 3, . . . 4 . We define a countable multinorm S s  a k : k g N0 4 on D by

a k Ž w . [ 5 T kw 5 ,

k g N0 ,

w g D.

DEFINITION 2.1. Ža. We denote by D`ŽT . the countably multinormed space consisting of the set D with topology generated by the countable multinorm S. Žb. We denote by D`X ŽT . the space of all continuous linear functionals on D`ŽT .. We impose the weak* topology on D`X ŽT .. Thus, sequential convergence in D`X ŽT . is given by f n ª f in D`X Ž T . as n ª ` m² f n , w: ª² f , w: as n ª ` ;w g D`Ž T . , where ² f , w: denotes the action of f g D`X ŽT . on w g D`ŽT ..

226

BAPTISTE, LAMB, AND MCGHEE

THEOREM 2.2.

D`ŽT . is a Frechet space. ´

Proof. This follows as in w6, Theorem 2.3x; we need note only that T normal implies that T k is closed for all k g N [  1, 2, 3, . . . 4 . THEOREM 2.3.

D`ŽT . is dense in H.

Proof. By w11, p. 364x we know that T s UP s PU where U is unitary, P is positive and self-adjoint, and DŽT . s DŽ P .. Thus, D Ž T k . s D Ž Ž UP .

k

. s DŽ U kP k . s DŽ P k .

;k g N0 ,

and the result follows as in w6, Theorem 2.4x. THEOREM 2.4. T k : D`ŽT . ª D`ŽT . is continuous for all k g N0 . Proof. This follows as in w6, Theorem 2.2x. THEOREM 2.5.

D`X ŽT . is sequentially complete.

Proof. This follows immediately from the completeness of D`ŽT . Žsee w13x.. THEOREM 2.6. Each h g H generates, uniquely, a continuous linear functional h ˜ g D`X ŽT ., ¨ ia the formula

² h˜ , w: [ Ž w , h . ,

w g D`Ž T . .

Proof. This follows as in w6, Theorem 2.7x. As a consequence of Theorem 2.6, we can regard H as a subspace of D`X ŽT . and we have the triple D`Ž T . ¨ H ¨ D`X Ž T . , where ¨ denotes a continuous embedding; the continuity of the embeddings is easily verified. We now turn our attention to the problem of obtaining&a continuous extension T˜ of T to D`X ŽT .. In particular, we require T˜h ˜ s Th ;h g DŽT ., that is, &

² T˜h˜ , w: s² Th , w: s Ž w , Th . s Ž T *w , h . s² h ˜ , T *w:

;w g D`Ž T . , ;h g D Ž T . .

Note that w g D`ŽT . « w g DŽT . s DŽT *. since T is a normal operator. ˜ Thus, we are led to the following definition of T.

EXTENSIONS OF NORMAL OPERATORS

227

DEFINITION 2.7. The extension of T to D`X ŽT ., denoted by T˜ : D`X ŽT . ª is defined by

D`X ŽT .,

; f g D`X Ž T . , w g D`Ž T . .

˜ , w: [² f , T *w: ² Tf

That is, T˜ [ w T *x9, the D`ŽT .-adjoint of the Hilbert-adjoint T * of T. This should be compared with the self-adjoint case w6x where we have T˜ [ T 9. LEMMA 2.8. T * g LŽ D`ŽT .., the space of all continuous linear operators on D`ŽT .. Proof. Since T is normal, it follows that T k is normal. Consequently, DŽT k . s DŽŽT k .*. s DŽŽT *. k . for all k g N, and it follows easily that T * maps D`ŽT . into D`ŽT .. Also, for all k g N,

a k Ž T *w . s 5 T k T *w 5 s 5 T *T kw 5

since T *T s TT *,

s 5 TT kw 5

since 5 T *c 5 s 5 Tc 5 ;c g D Ž T .

s a kq 1 Ž w . , and it follows that T * is continuous on D`ŽT .. THEOREM 2.9. T˜ g LŽ D`X ŽT ... Proof. This follows immediately from Definition 2.7, Lemma 2.8, and w2, p. 65x. & Defining T k, the extension of T k , via &

¦T

k

f , w [² f , Ž T k . *w:

;

; f g D`X Ž T . , w g D`Ž T . ,

we obtain the following result. &

COROLLARY 2.10. T ks T˜ k ;k g N. Proof. &

¦T

k

f , w [ ² f , Ž T k . *w:

;

k

s ² f , Ž T * . w: s ² T˜ k f , w:

; f g D`X Ž T . , w g D`Ž T . , k g N.

An exact characterisation of the elements of D`X ŽT . is given by the following.

228

BAPTISTE, LAMB, AND MCGHEE

THEOREM 2.11. f g D`X ŽT . if and only if there exist a non-negati¨ e integer n n and a sequence Žhr . rs0 ; H Ž both dependent upon f . such that n

Ý T˜rh˜r .

fs

rs0

Proof. Sufficiency. It follows from Theorems 2.6 and 2.9 and Corollary 2.10 that

hr g H « h ˜r g D`X Ž T . « T˜rh˜r g D`X Ž T .

; r s 0, . . . , n,

and hence n

f[

Ý T˜rh˜r g D`X Ž T . . rs0

Necessity. Let f g D`X ŽT .. Then, by w13, Lemma 1.10-1x, 'c ) 0 and an integer n g N0 such that

² f , w: F c max  5 T rw 5 4

;w g D`Ž T . .

0FrFn

Ž 2.1.

Let H

nq 1

1 times ! n q# "

[ H=...=H

with an inner product defined by n

Ž F, C. [

Ý Ž wi , ci . , is0

where F s Ž w 0 , w 1 , . . . , wn . g H nq 1 ,

C s Ž c 0 , c 1 , . . . , cn . g H nq 1 .

Let P : D`ŽT . ª H nq 1 be defined by n

2

Pw [ Ž w , T *w , Ž T * . w , . . . , Ž T * . w . ,

w g D`Ž T . .

Clearly, Pw s Pc m w s c , so that P is injective. Define Q on P Ž D`ŽT .. by Q : P Ž D`Ž T . . ª C 2

n

Q Ž w , T *w , Ž T * . w , . . . , Ž T * . w . [ ² f , w: .

229

EXTENSIONS OF NORMAL OPERATORS

Then, by Ž2.1., n

< Q Ž w , T *w , Ž T * . 2 w , . . . , Ž T * . w . < s < ² f , w: < F c max  5 T rw 5 4 0FrFn r

s c max  5 Ž T * . w 5 4 F c 0FrFn

n

Ý

1r2 r

5 Ž T *. w 5 2

s c 5 Pw 5 ,

rs0

so that Q is a continuous linear functional on P Ž D`ŽT ... By the Hahn]Banach theorem, we can extend Q to a continuous linear functional on H nq 1 ; we continue to denote this extension by Q. Now, by the Riesz representation theorem, there exists h s Žh 0 , h1 , . . . , hn . g H nq 1 such that n

QŽ F . s Ž F , h . s

Ý Ž wr , hr .

; F g H nq 1 .

rs0

Hence, n

² f , w: s Q Ž w , T *w , Ž T * . 2 w , . . . , Ž T *. n w . s Ý Ž Ž T *. r w , hr . rs0

s

n

n

rs0

rs0

&

r Ý ² h˜r , Ž T *. w: s Ý ² T rh˜r , w:

&

;w g D`Ž T . ,

so that f s Ý nrs0T rh ˜r , as required. 3. SPECTRAL THEORY IN D`ŽT . AND D`X ŽT . A standard result in spectral theory is that each w g H can be represented in the form

ws

H dE Ž z . w ,

Ž 3.1.

where E is the projection-valued spectral measure associated with the normal operator T, and similarly, T rw s

Hz

r

dE Ž z . w

;w g D Ž T r . , r g N.

Here and throughout, spectral integrals are over all of C. For every w g H, Ew Ž ? . [ 5 E Ž ? . w 5 2

Ž 3.2.

230

BAPTISTE, LAMB, AND MCGHEE

defines a real-valued measure on the Borel subsets of C, and a function is said to be E-measurable if it is Ew-measurable for all w g H. For a given E-measurable function F, an operator F ŽT . may be defined by DŽ F Ž T . . [ w g H :

½

FŽT . w [

H< F Ž z . < dE Ž z . - ` 5 2

w

Ž 3.3.

HF Ž z . dE Ž z . w .

The vector-valued integrals in Ž3.1., Ž3.2., and Ž3.3. all exist strongly in H, so that in Ž3.3., for example, -lim H s Ž z . dE Ž z . w H F Ž z . dE Ž z . w [ H nª` n

for any sequence of step functions Ž sn .`ns1 such that sn ª F in L2 ŽC, Ew ., the complex L2-space of functions that are absolutely square integrable on C with respect to the measure Ew . For details of this development of the spectral theory for normal operators, we refer to w12x. Our aim in this section is to obtain generalisations of Ž3.1. ] Ž3.3. to the spaces D`ŽT . and D`X ŽT .. In particular, we must show that spectral integrals can be well defined in D`ŽT . and D`X ŽT .. We begin with the space D`ŽT . and examine the complex spectral measure E. LEMMA 3.1.

For e¨ ery Borel set J ; C, EŽ J . g LŽ D`ŽT ...

Proof. For every Borel set J ; C, EŽ J . g LŽ H . and is a self-adjoint projection operator. Therefore, for any w g D`ŽT .,

a k Ž E Ž J . w . s 5 T k E Ž J . w 5 s 5 E Ž J . T kw 5 F 5 T kw 5 s a k Ž w . from which the continuity of EŽ J . on D`ŽT . follows. We now define integration in D`ŽT .. For a step function n

sŽ z . s

Ý cj X J Ž z . j

js1

and w g D`ŽT ., we define n

D`Ž T . -

H

s Ž z . dE Ž z . w [

Ý c j E Ž Jj . w . js1

231

EXTENSIONS OF NORMAL OPERATORS

Then, for a bounded E-measurable function F and any w g D`ŽT ., there exists a bounded sequence of step functions Ž sn .`ns1 Žsee w12, Auxiliary Theorem 7.13x., Nn

sn Ž z . s

Ý cn j X J Ž z . nj

js1

such that sn ª F

in L2 Ž C, Ew .

« sn ª F

a.e. with respect to Ew Ž since sn and F are bounded.

« sn ª F

a.e. with respect to ET kw ;k g N;

the last implication follows since, for any Borel set J ; C, and any w g D`ŽT ., Ew Ž J . s 5 E Ž J . w 5 2 s 0 « ET kw Ž J . s 5 E Ž J . T kw 5 2 s 5 T k E Ž J . w 5 2 s 0. Using the Legesgue dominated convergence theorem we deduce that sn ª F

in L2 Ž C, ET kw . ;k g N.

Ž 3.4.

Since F is a bounded function,

w g D`Ž T . « F Ž T . w g D`Ž T . and we have, for all k g N0 ,

ak F Ž T . w y

ž

H s Ž z . dE Ž z . w / n

ž

Nn

s T k FŽT . w y s 5 F Ž T . T kw y

Ý c n j E Ž Jn j . w js1

/

Nn

Ý c n j E Ž Jn j . T kw 5 js1

s 5 F Ž T . T kw y

H s Ž z . dE Ž z . T w 5 ª 0 k

n

as n ª `

by Ž3.4.. Therefore, for a bounded E-measurable function F, and any w g D`ŽT ., we define D`Ž T . -

-lim H s Ž z . dE Ž z . w s F Ž T . w . H F Ž z . dE Ž z . w [ D Ž T . nª` `

n

232

BAPTISTE, LAMB, AND MCGHEE

We now consider spectral integrals of unbounded functions. The next theorem gives necessary and sufficient conditions on F for the operator F ŽT . defined in H by Ž3.3. to be a continuous operator on D`ŽT .. THEOREM 3.2. Let F be an E-measurable function defined on C. Then, F ŽT . defined by Ž3.3. is a continuous operator on D`ŽT . if and only if there exist m g N0 and M - ` such that sup < F Ž z . < Ž 1 q < z < 2 .

ym

zg s ŽT .

s M,

Ž 3.5.

that is, if and only if F ŽT .Ž I q T *T .ym g LŽ H ., where I denotes the identity operator. Proof. Sufficiency. Suppose F satisfies Ž3.5.. Then, for any w g D`ŽT . and any k g N0 ,

H
k

F Ž z . < 2 dEw Ž z . s

H Ž1 q < z <

F M2

2 y2 m

.

H < Ž1 q < z <

2

Ž1 q < z < 2 .

m

.

2m

< z < 2 k < F Ž z . < 2 dEw Ž z .

z k < 2 dEw Ž z . m

s M 2 5 T k Ž I q T *T . w 5 2 - `. Hence, defining Gk Ž z . [ z k , it follows that

w g D Ž Ž FGk . Ž T . .

;w g D`Ž T . , k g N0 .

By w11, Theorem 13.24Žb.x we deduce that

w g D Ž F Ž T . T k . l D ŽT kF Ž T . .

;k g N0 ,

and, for each k g N0 , we have m

5 T k F Ž T . w 5 F M 5 T k Ž I q T *T . w 5 . Thus, for all w g D`ŽT ., m

a k Ž F Ž T . w . s 5 T k F Ž T . w 5 F M a k Ž Ž I q T *T . w . . Since Ž I q T *T . m g LŽ D`ŽT .. for each m g N0 , it follows that F ŽT . g LŽ D`ŽT ... Necessity. Now suppose that F ŽT . g LŽ D`ŽT ... Then, by w13, Lemma 1.10-1x, there exist c ) 0 and m g N0 such that 5 F Ž T . w 5 2 s a0 Ž F Ž T . w .

2

F c 2 max

0FjFm

½

aj Ž w .

2

5

;w g D`Ž T . .

233

EXTENSIONS OF NORMAL OPERATORS

Hence, 5 F Ž T . Ž I q T *T .

ym

w 5 2 F c 2 max 5 T j Ž I q T *T .

ym

0FjFm

s c 2 max

0FjFm

F c2

H dE

w

½H

< z j Ž1 q < z < 2 .

w52

ym

< 2 dEw

5

s c2 5 w 5 2 .

Thus, F ŽT .Ž I q T *T .ym g LŽ H . and so Ž3.5. holds. Now, for any E-measurable function F satisfying Ž3.5., we define cut-off functions Fn by Fn Ž z . [

½

if < F Ž z . < - n, otherwise,

FŽ z. 0

Ž 3.6.

and we define D`Ž T . -

-lim H F Ž z . dE Ž z . w H F Ž z . dE Ž z . w [ D Ž T . nª` `

n

Ž 3.7.

s D`Ž T . -lim Fn Ž T . w . nª`

THEOREM 3.3. Gi¨ en an E-measurable function F : C ª C satisfying Ž3.5. and cut-off functions Fn defined by Ž3.6., then Ž3.7. makes sense, and D`Ž T . -

H F Ž z . dE Ž z . w s F Ž T . w .

Proof. For any w g D`ŽT .,

a k Ž F Ž T . w y Fn Ž T . w . s 5 T k Ž F Ž T . w y Fn Ž T . w . 5 s 5 F Ž T . T kw y Fn Ž T . T kw 5 ª 0

as n ª `

since, for all w g D`ŽT . and k g N0 , F Ž T . T kw s

-lim H F Ž z . dE Ž z . T w H F Ž z . dE Ž z . T w s H nª` k

k

n

s H -lim Fn Ž T . T kw . nª`

234

BAPTISTE, LAMB, AND MCGHEE

Thus, Fn Ž T . w ª F Ž T . w

in D`Ž T . .

Note that, in all cases, the D` -integral and the H-integral are identical as elements in H, and so we can usually omit the prefix on the integrals in D`ŽT . and H without loss of understanding. It is now straightforward to show that we have a good functional calculus in D`ŽT .. THEOREM 3.4. Let F and G be E-measurable functions on C which satisfy Ž3.5. for suitable ¨ alues of m and M. Then, Ža. Ž a F q b G .ŽT . s a F ŽT . q b GŽT . g LŽ D`ŽT .. ;a , b g C; Žb. Ž FG .ŽT . s F ŽT .GŽT . g LŽ D`ŽT ... Proof. The results follow as in w6, Theorem 3.6x. We now turn our attention to the space D`X ŽT .. Firstly, we extend the spectral measure E to D`X ŽT .. DEFINITION 3.5. For any Borel set J ; C the extension of EŽ J . to D`X ŽT ., denoted by E˜Ž J . : D`X ŽT . ª D`X ŽT ., is defined by

² E˜Ž J . f , w: [ ² f , E Ž J . w: ,

f g D`X Ž T . , w g D`Ž T . .

It is clear that E˜ is an operator-valued measure defined on the Borel subsets of C. Following our development of spectral integrals in D`ŽT ., we now consider integrals in D`X ŽT .. For a step function sŽ z . s Ý njs1 c j X J jŽ z . we define H sŽ z . dE˜Ž z . : D`X ŽT . ª D`X ŽT . by

¦H Ž .

;

n

s z dE˜Ž z . f , w [

¦Ý

js1

¦

s f,

;¦

c j E˜Ž J j . f , w s

n

f,

;

Ý c j E Ž Jj . w js1

H s Ž z . dE Ž z . w;s ² f , s Ž T . w: , f g D`X Ž T . , w g D`Ž T . .

For a bounded E-measurable function F and any w g D`ŽT ., we have, as before, a sequence of step functions Ž sn .`ns1 converging to F in

EXTENSIONS OF NORMAL OPERATORS

235

L2 ŽC, ET kw . and a.e. with respect to ET kw for all k g N0 . We define H F Ž z . dE˜Ž z . : D`X ŽT . ª D`X ŽT . by

¦H

;

F Ž z . dE˜Ž z . f , w

¦H Ž . ˜Ž . ; ¦ H Ž. Ž.; ¦ H Ž. Ž.; ¦ H Ž. Ž.; ² sn z dE z f , w

[ lim

nª`

f,

sn z dE z w

s f , lim

sn z dE z w

s lim

nª`

nª`

Ž f g D`X Ž T . , w g D`Ž T . .

by the continuity of f

F z dE z w s f , F Ž T . w: .

s f,

Finally, for an unbounded E-measurable function F satisfying Ž3.5., we use the cut-off functions Fn given in Ž3.6. and define H F Ž z . dE˜Ž z . : D`X ŽT . ª D`X ŽT . by

¦H

;

F Ž z . dE˜Ž z . f , w [ lim

nª`

¦H

;

Fn Ž z . dE˜Ž z . f , w

Ž f g D`X Ž T . , w g D`Ž T . .

s lim ² f , Fn Ž T . w: nª`

s f , lim Fn Ž T . w

¦

nª`

;

¦

s ² f , F Ž T . w: s f ,

H F Ž z . dE Ž z . w;

since f is continuous and FnŽT . w ª F ŽT . w in D`ŽT .. We should note that integrals in D`X ŽT . are defined in a pointwise fashion, that is, ² H F Ž z . dE˜Ž z . f , w: is defined for each pair Ž f, w . g D`X ŽT . = D`ŽT .. We can now define a functional calculus in LŽ D`X ŽT ..; this, of course, must be consistent with Definition 2.7. DEFINITION 3.6. For any E-measurable function F : C ª C satisfying Ž3.5., we define F Ž T˜ . [

H F Ž z . dE˜Ž z . .

236

BAPTISTE, LAMB, AND MCGHEE

Thus,

² F Ž T˜. f , w: s

¦H Ž . ˜Ž . ; ¦ H Ž. Ž.; F z dE z f , w

s f,

F z dE z w s ² f , F Ž T * . w: ,

f g D`X Ž T . , w g D`Ž T . ,

that is, F Ž T˜ . s Ž F Ž T * . . 9,

Ž 3.8.

the D`ŽT .-adjoint of F ŽT *. and, as such, F ŽT˜. is a continuous operator on D`X ŽT .. In particular, T˜ s

H zdE˜Ž z . s Ž T *. 9

in agreement with Definition 2.7. That we have a good functional calculus in LŽ D`X ŽT .. is shown by the following: THEOREM 3.7. Let F and G be E-measurable functions on C which satisfy Ž3.5. for suitable ¨ alues of m and M. Then, Ža. Ž a F q b G .ŽT˜. s a F ŽT˜. q b GŽT˜. g LŽ D`X ŽT .. ;a , b g C; Žb. Ž FG .ŽT˜. s F ŽT˜.GŽT˜. g LŽ D`X ŽT ... Proof. These results are straightforward calculations from Ž3.8.. Also of interest is the extended version of F ŽT .. In accordance with & Definition 2.7, F Ž T . is defined on D`X ŽT . via &

² F Ž T . f , w: [ ² f ,

F Ž T . *w:

; f g D`X Ž T . , w g D`Ž T . . Ž 3.9.

Since w F ŽT .x* s F ŽT . and F satisfies Ž3.5. whenever F does, we can immediately state the following result. THEOREM 3.8. Let F be any E-measurable function on C satisfying Ž3.5.. & Then F Ž T . g LŽ D`X ŽT ... Proof. This is obvious. &

We note that F Ž T . s F ŽT˜. m F Ž z . s F Ž z ..

237

EXTENSIONS OF NORMAL OPERATORS

4. APPLICATIONS To illustrate the preceding theory, we consider the Hilbert space L2m Ž 0, ` . s  w : xymw Ž x . g L2 Ž 0, ` . 4 ,

m g R,

with inner product and norm `

Ž w , c . m s H xy2 mw Ž x . c Ž x . dx, 0

5w5m s Ž w, c . m

1r2

.

By choosing a suitable normal realisation on L2mŽ0, `. of the differential operator D defined by

Ž Dw . Ž x . [ x w 9 Ž x . ,

x ) 0,

we shall demonstrate that the corresponding space D`Ž D . coincides with the test function space F2, m used extensively in the work of McBride Žsee, for example, w7x.. This space can be defined in several different but equivalent ways. For our purposes, the most convenient definition is the following slight modification of w9, Definition 2.5x: F2, m [  w g C` Ž 0, ` . : D kw g L2m Ž 0, ` . ;k g N0 4 . Equipped with the topology generated by the countable multinorm g km : k g N0 4 where

g km Ž w . [ 5 D kw 5 m ,

w g F2, m ,

F2, m can be shown to be a Frechet space which meets the requirements of ´ a test function space stipulated in w13x. A natural choice for the domain of the operator D is the maximal domain D Ž D . [  w g L2m Ž 0, ` . : w is absolutely continuous on Ž 0, ` . and Dw g L2m Ž 0, ` . 4 . To show that this maximal operator is normal on L2mŽ0, `., we require the following two lemmas. LEMMA 4.1.

Let w , c g DŽ D .. Then

lim x 1y 2 mw Ž x . c Ž x . s lim x 1y2 mw Ž x . c Ž x . s 0.

xª0q

xª`

238

BAPTISTE, LAMB, AND MCGHEE

Proof. Firstly, note that x 1y 2 mw Ž x .c Ž x . is absolutely continuous on Ž0, `. and that xy2 mw Ž x .c Ž x . g L1 Ž0, `.. Now `

H0

d

x 1y 2 mw Ž x . c Ž x . dx

dx s

`

H0

xy2 m Ž 1 y 2 m . w Ž x . c Ž x . q Ž Dw . Ž x . c Ž x . q w Ž x . Ž Dc . Ž x . dx

and the latter integral converges since w , c , Dw , and Dc are all in L2mŽ0, `.. Consequently, there exist finite non-negative constants M and N such that lim xª0q

x 1y 2 mw Ž x . c Ž x . s M,

lim x 1y2 mw Ž x . c Ž x . s N.

xª`

If M / 0 then there exist e g Ž0, M . and d ) 0 such that < x 1y 2 mw Ž x . c Ž x . < y M - e « xy2 mw Ž x . c Ž x . )

Mye

; x g Ž 0, d . ; x g Ž 0, d .

x

« xy2 mw Ž x . c Ž x . f L1 Ž 0, ` . . Since this is a contradiction, we deduce that M s 0. A similar argument shows that N s 0. LEMMA 4.2. The maximal operator D has adjoint D* gi¨ en by D Ž D* . s D Ž D . D*w s Ž 2 m y 1 . I y D w ,

w g DŽ D. .

Proof. Let w , c g DŽ D .. Then `

Ž Dw , c . m s H x 1y 2 mw 9 Ž x . c Ž x . dx 0

s x 1y2 mw Ž x . c Ž x . sy

`

H0

` 0

y

`

H0

wŽ x.

d dx

x 1y2 m c Ž x . dx

w Ž x . Ž 1 y 2 m . xy2 m c Ž x . q x 1y2 m c 9 Ž x . dx

Ž by Lemma 4.1. s Ž w , Ž 2 m y 1. I y D c . m , and we deduce that Ž2 m y 1. I y D ; D*.

239

EXTENSIONS OF NORMAL OPERATORS

Now, let w g DŽ D ., c g DŽ D*., and let c * s D*c . Then `

H0

x 1y 2 mw 9 Ž x . c Ž x . dx s

`

H0

xy2 mw Ž x . c * Ž x . dx.

Ž 4.1.

Let wnŽ?; y, z . Ž n g N, 0 - y - z . be the continuous function that satisfies

wn Ž x ; y, z . s 1

for x g w y, z x ;

wn Ž x ; y, z . s 0

for x F y y

1 n

or x G z q n1 ;

wn Ž ?; y, z . is linear on y y n1 , y and on z, z q

1 n

.

Then wnŽ?; y, z . g DŽ D . and from Ž4.1. we obtain y

n

Hyy1rn x s

1y 2 m

`

H0

c Ž x . dx y n

Hz

zq1rn 1y2 m

x

c Ž x . dx

xy2 mwn Ž x ; y, z . c * Ž x . dx.

Ž 4.2.

Letting n ª ` in Ž4.2. and taking conjugates gives y 1y 2 mc Ž y . y z 1y2 mc Ž z . s

z

y2 m

Hy x

c * Ž x . dx,

and since xy2 mc *Ž x . is integrable over any finite subinterval of Ž0, `., it follows that z 1y 2 mc Ž z . is an absolutely continuous function of z on Ž0, `.. Moreover, zy2 mc * Ž z . s y

d dz

z 1y2 mc Ž z . s zy2 m Ž 2 m y 1 . I y D c Ž z . ,

which shows that

c g DŽ D.

and

c * s D*c s Ž 2 m y 1 . I y D c ;

we deduce that D* ; Ž2 m y 1. I y D. This completes the proof. THEOREM 4.3. The maximal operator D is normal on L2mŽ0, `.. Proof. This follows directly from Lemma 4.2.

240

BAPTISTE, LAMB, AND MCGHEE

D`Ž D . s F2, m .

THEOREM 4.4.

Proof. Let w g D`Ž D .. Then D kw is absolutely continuous on Ž 0, ` . for each k g N0 « D kw is continuous on Ž 0, ` . for each k g N0 « w g C` Ž 0, ` . . Also, D kw g L2mŽ0, `. for each k g N0 and

a k Ž w . s g km Ž w . s 5 D kw 5 m

;w g D`Ž D . .

Hence, D`Ž D . ¨ F2, m . The reverse embedding F2, m ¨ D`Ž D . is obvious. Theorem 4.4 has several immediate consequences. Firstly, the generalized function spaces F2,X m and D`X Ž D . must also be identical. Secondly, the spectral theory developed in Section 3 can now be used to produce a functional calculus for the operator D on F2, m and also for the distribu˜ on F2,X m . Indeed, the fact that F2,X m s D`X Ž D . means tional extension D X that F2, m may be viewed as a space of generalized functions tailored to the & ˜ . or F Ž D . for suitably measurable needs of operators of the form F Ž D functions F satisfying sup < F Ž z . < Ž 1 q < z < 2 .

ym

zg s Ž D .

s M,

m g N0 , M - `.

This in turn leads to the interesting possibility of identifying specific functions F which will produce results in F2,X m for extensions of classical integral and differential operators. As a simple illustration, we consider the logarithmic fractional integral, or Hadamard operator w4, p. 672x, H a defined by

Ž H aw . Ž x . [

1

x

H GŽ a . 0

x

ay1

yy1w Ž y . dy,

ž / log

y

x ) 0, Re a ) 0.

Ž 4.3. Direct application of an inequality due to Hardy w3, Theorem 319x, establishes that 5 H w5m F a

G Ž Re a .
ž

my

1 2

yR e a

/

5w5m

;w g L2m Ž 0, ` .

Ž 4.4.

241

EXTENSIONS OF NORMAL OPERATORS

when Re a ) 0 and m ) 1r2. Our aim is to verify that, under the same conditions on a and m , H a s Dya as an operator on F2, m . We require the following spectral properties of D on L2mŽ0, `., each of which can be obtained by routine calculation which we omit. THEOREM 4.5. Ža. s Ž D . s  z g C : Re z s m y 1r24 . Žb. For Re l ) 1r2 y m , Ž D q l I .y1 s my lH 1 m l, where m denotes the operation of multiplication by the ¨ ariable. Let Fa Ž z . [ z a s exp Ž a Log z . ,

a g C,

where Log z denotes the principal branch of the complex logarithm. If m ) 1r2, then it is clear from Theorem 4.5 that Fa Ž z . is well defined for each z g s Ž D ., and also there exist constants ma g N0 and Ma - ` such that sup < Fa Ž z . < Ž 1 q < z < 2 .

ym a

zg s Ž D .

s Ma .

Consequently, from Theorems 3.2 and 3.8, D a [ Fa Ž D . g L Ž F2, m .

and

&

&

D a [ Fa Ž D . g L Ž F2,X m .

for each a g C provided that m ) 1r2. Now suppose that Re a g Ž0, 1. and consider Dya on F2, m . Then, for w g F2, m , Dyaw s s s s

HF

ya

Ž z . dE Ž z . w

sin pa

p sin pa

p sin pa

p

½

`

H H0 `

H0

`

H0

sya

sya sqz

½H

5

ds dE Ž z . w 1

sqz

sy a Ž D q sI .

Ž by

1, p310 Ž 19 .

.

5

dE Ž z . w ds y1

w ds.

Ž 4.5.

Since, by Ž4.4.,

g km Ž Ž D q sI .

y1

w . s 5 Ž D q sI .

y1

D kw 5 m F m q s y

ž

1 2

y1

/

g km Ž w . , ;w g F2, m ,

242

BAPTISTE, LAMB, AND MCGHEE

the integral in Ž4.5. converges in F2, m . The interchange of the order of integration in the above calculation can be justified by using the Fubini]Tonelli Theorem Žsee w12, p. 374x.. Therefore, we can use the continuity on F2, m of the translated delta distribution d x , given by

² d x , w: [ w Ž x . ,

x ) 0,

to obtain

² dx ,

Dyaw: s s s s

sin pa

p sin pa

p sin pa

p 1 GŽ a

`

H0

sy a

x

H0

x

ys

x

yy1w Ž y . dyds

ž / y

y1

`

½

H0 w Ž y . y H0 x

H0 w Ž y .

x

ž / H ž / . x

log

0

x

log

y

sy a exp ys log

ay1

yy1

ž ½H

`

x y

/ 5 5

ds dy

uy a eyudu dy

0

ay1

yy1w Ž y . dy s Ž H aw . Ž x . ,

y

; x ) 0.

Hence Dyaw s H aw

;w g F2 , m , 0 - Re a - 1.

Similarly, when 1 F Re a - 2, we can write a s a 1 q a 2 , where Re a i g Ž0, 1., i s 1, 2, and use standard index laws associated with H a to deduce that Dyaw s Dya 1 Dya 2 w s H a 1 H a 2 w s H a 1qa 2 w s H aw

;w g F2, m .

Continuing in this way produces the result Dyaw s H aw

;w g F2, m , Re a ) 0.

Ž 4.6.

Since the left-hand side of Ž4.6. is well defined for any a g C, we have a means of extending the fractional integral semigroup  H a : Re a ) 04 ; LŽ F2, m . to a group of operators  H a : a g C4 ; LŽ F2, m . where, for Re a F 0, H aw [ Dyaw ,

w g F2, m .

Using the functional calculus in D`Ž D ., it is easily seen that, as operators on F2, m , H aH b s H aq b ,

;a , b g C,

H s I, 0

ŽH a.

y1

s H ya ,

;a g C.

243

EXTENSIONS OF NORMAL OPERATORS

Also, when n y 1 - Re a - n, a concrete form of H y a is given by H y a s H ny a D n . A distributional theory for operators H a can now be produced almost immediately. Since s as s a , it follows from Ž3.9. that &

&

² H a f , w: s ² Dy a f , w: ya [ ² f , Ž D* . w: s ² f , Ž 2 m y 1. I y D

ya

w: ,

; f g F2,X m , w g F2, m , a g C.

By arguing as before, we can show that, for w g F2, m ,

Ž 2 m y 1. I y D

ya

wŽ x. s

1

`

H G Ž a q n. x

x

2 m y1

y

= yy1 Ž 2 m y 1 . I y D &

y

a qny1

ž / ž / log

n

x

w Ž y . dy

&

whenever Re a q n ) 0. Note also that H a[ Fa Ž D . is identical to the ˜ . on F2,X m and therefore the functional calculus in D`X Ž D . ŽD operator Fa& shows that H a has the same properties on F2,X m as H a on F2, m . Further investigations indicate that a variety of Mellin multiplier transforms, including, for example, the Erdelyi]Kober operators of fractional ´ calculus, can be treated in a similar manner. It is our intention to pursue this in a future paper.

REFERENCES 1. A. Erdelyi ´ et al., ‘‘Tables of Integral Transforms,’’ Vol. 1, McGraw]Hill, New York, 1954. 2. I. M. Gel’fand and G. E. Shilov, ‘‘Generalized Functions,’’ Vol. 2, Academic Press, New York, 1968. 3. G. H. Hardy, J. E. Littlewood, and G. Polya, ‘‘Inequalities,’’ Cambridge Univ. Press, ´ Cambridge, 1934. 4. E. Hille and R. S. Phillips, ‘‘Functional Analysis and Semigroups,’’ Amer. Math. Soc. Colloq. Publ., Amer. Math. Soc., Providence, RI, 1957. 5. D. Judge, On Zemanian’s distributional eigenfunction transforms, J. Math. Anal. Appl. 34 Ž1971., 187]201. 6. W. Lamb and D. F. McGhee, Spectral theory and functional calculus on spaces of generalized functions, J. Math. Anal. Appl. 163 Ž1992., 238]260. 7. A. C. McBride, ‘‘Fractional Calculus and Integral Transforms of Generalized Functions,’’ Pitman Research Notes in Mathematics, Vol. 31, Pitman, London, 1979. 8. A. C. McBride, Fractional powers of a class of Mellin multiplier transforms, I, Appl. Anal. 21 Ž1986., 89]127; II, 129]149; III, 151]173.

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BAPTISTE, LAMB, AND MCGHEE

9. A. C. McBride, Connections between fractional calculus and some Mellin multiplier transforms, in ‘‘Univalent Functions, Fractional Calculus, and Their Applications’’ ŽH. M. Srivastava and S. Owa, Eds.., Ellis Horwood, Chichester, 1989. 10. D. F. McGhee, F. Baptiste, and W. Lamb, A distributional version of spectral theory for normal operators, in ‘‘Evolution Equations’’ ŽG. Ferreyra, G. Goldstein, and F. Neubrander, Eds.., Dekker, New York, 1994. 11. W. Rudin, ‘‘Functional Analysis,’’ McGraw]Hill, New Delhi, 1973. 12. J. Weidmann, ‘‘Linear Operators in Hilbert Spaces,’’ Springer, New York, 1980. 13. A. Zemanian, ‘‘Generalized Integral Transformations,’’ Interscience, New York, 1968.