A theory of generalized functions based on holomorphic semi-groups: part C : linear mappings, tensor products and kernel theorems

A theory of generalized functions based on holomorphic

semi-groups

Citation for published version (APA):

Graaf, de, J. (1983). A theory of generalized functions based on holomorphic semi-groups: part C : linear mappings, tensor products and kernel theorems. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8308). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1983

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TECHNISCHE HOGESCHOOL EINDHOVEN NEDERLAND

ONDERAFDELING DER WISKUNDE

TECHNOLOGICAL UNIVERSITY EINDHOVEN THE NETHERLANDS

DEPARTMENT OF MATHEMATICS

Memorandum 83-08

J. de Graaf

A THEORY OF GENERALIZED FUNCTIONS BASED ON HOLOMORPHIC SEMI-GROUPS

CHAPTER 4. Characterization of continuous linear mappings between the spaces

SX,A' TX,A' SV,S

and

TV,S

C.l

Let

S

be a nonnegative self-adjoint operator in a separable Hilbert space

V.

As before

A

is a nonnegative self-adjoint operator in a separable Hilbert space

X.

In this chapter we shall derive conditions implying the continuity of linear mappings

SX,A

~

SV,S' SX,A

~

TV,S' TXtA

~

TV,S' TX,A

~

SV,S'

Further we investigate which linear operators, defined on a,subset of

X,

can be continously extended to operators on

TX,A'

The next theorem is an immedi-ate consequence of the fact that

SX,A

is bomological. Cf. Theorem 1.11. For completeness we give an ad hoc proof.

Theorem 4. 1. A linear map

Q :

Sx ,A ~ V I where V is an arbi trary locally convex topological vector space, is continuous

I. iff for each t > a the map Qe -tA : X ~ V is continuous i

II. iff for each null sequence {un} c

SX,A'

un + a in

SX,A'

the sequence

{Qu } is a null sequence in

V.

n

Proof

I. • ) e -tA . ~s an isomorphism from

X

to

X

t" By the definition of the induc-tive limit,

X

is continuously injected in

SX,A'

SO if Q is continuous it

t -tA

follows that Qe is con tinuous •

.. ) Let ~ denote the restriction of

Q

to X

t' From the continuity of -tA

Qe on

X

follows the continuity of ~ on

X

_t• Now let 0 3 a be open in

V.

For each t > 0, -1 X t• Thus

Q

en)

is -1 -1

Q

(0) n

X

t

=

~ (0) is an open a-neighbourhood in open in

SX,A'

II. Follows from I because null sequences in

S

are always null sequences in some

X

_t' t > 0, and vice versa.

In the next theorem we characterize continuous linear mappings from

SX,A

to

Sy,S"

Theorem 4.2. Suppose P : SX,A .... SY,B is a linear mapping. The following seven condi tions are equi valen t.

C.2

I. P is continuous with respect to the strong topologies of SX,A and Sy,B' II. un ....

a,

strongly in SX,A' implies Pun ..,. 0, strongly in SY,B'

III. For each a > 0 the operator Pe

-aA

is a bounded linear operato.r from

X

into y,

IV. For each a >

a

and each W € B the opera tor W (B) Pe -aA is a bounded

linear operator from

X

into Y.

V. For each t > 0 there exists 6 >

a

such tha t Pe

-tA

(X) c e -6B (Y) and e 6B Pe

-tA

is a bounded linear opera tor from X in to Y.

VI. There exists a dense linear subspace

=

c Y such that for each fixed y €

=

the linear functional Lp,y(f)

=

(Pf,y)y is continuous on SX,A'

VII. For each t >

a

(Pe -tAl * is a bounded linear operator from Y to X. (Remark: One has (Pe-tA)*(y) c SX,A)'

Proof.

I ~ II. See Theorem 4.1.

II - III. If {x } is a null sequence in X then for any a >

a

n

-tA

null sequence in SX,A' By II {Pe x

n} is a null sequence in in

Y.

-aA

{e x } is a

n

Sy,B and hence

III - IV, The operator w(B)pe-aA is closed and defined on the whole of X. Therefore i t is bounded.

IV - V. Let U denote the unit ball in X. Because of IV the set Pe-aA(u) is bounded in SY,B' Now apply Theorem 1.6.

III - VI. We can take

=

= Y. Take y € Y fixed. For each x € X and each t > 0

we have

C.3

VI ... VII. According to Riess' theorem for each y e: E and each t ,. 0 there exis ts f t e: X such that for each h e: X

-TA

-TA

Replacing h by e x, x e: X, we observe that ft+T = e f_t so that f_t e: SX,A'

-tA

*

-tA

*

From (*) we obtain f

t = (Pe -1 y. SO DC (Pa » : : > E which is dense in

Y.

Since Pe -tA is defined on the whole X the operator CPe

-t~

*

is defined on

the whole Y and bounded, Repeating the argument with arbitrary y e X shows

-tA

*

(Pe ) y e SX,A'

-tA -tA

*

VII -to III. Pe is bounded because (Pe ) is bounded.

V .. II. Trivial.

The next corollary is important for applications.

Corollary 4.3. Suppose

Q

is a densely defined closable operator: X +

Y.

If

D(Q) ::> SX,A and Q(SX,A) c:

Sy,S'

then Qmaps SX,A continuously into

Sy,S'

*

-tA

Proof. Q is closable iff D (Q) is dense in Y. Since Qe : X + Y is defined on the whole X, its adjoint (Qe-tA)* is bounded. The adjoint, however, is

.

*

-tA

*

-tA

*

-tA

*

densely defined since on D (Q) one has (Qe )

=

e

Q •

Bence (Qe ) is defined on the whole

Y

and bounded. From this the boundedness of Qe-

tA

follows. Application of Theorem 4.2 III yields the desired result.

Theorem 4.4. Let K : SX,A + T Y ,S be a linear mapping, The following three condi tions are equi valen t.

I.

K

is continuous with respect to the strong topologies of SX,A and

Ty,S'

II. For each t > 0, a > 0, e -tB K e -aA is a bounded linear opera tor from X into

y.

III. For each t > 0, e -tB

K

is a continuous map from SX,A into

Sy,S'

Proof.

I -to II. Let {x } be a null-sequence in X.

n

in SX,A' Since

K

_{is continuous; {Ke-aAxn }}

Then {e -a Ax } is a null sequence n

is a null sequence in

Ty,S'

which

o

C.4

me-ans that for every t > 0 e-YS Ke-aA is bounded.

{(e-tBKe-aA)x} is a null sequence in

Y.

Hence n

II .. I. Let {un} be a null-sequence in SX.A' Then for defined and is a null-sequence in X.Butthen {Ku }

=

n

null-sequence in Ty,S since for each t > 0

. - $ II • III. Apply Theorem 4.2 with

P

= e

K.

in V •

Theorem 4.5. Let r TX,A ~ SV,S be a linear mapping. Let fr : X ~

Y

denote the restriction of r to X. The following five conditions are equivalent.

I. r is continuous with respect to the strong topologies of TX,A and Sy,S' * _{y ~ X is a bounded r* (V)}

c: SX,A'

I I . r opera tor and

r r

III. There exis ts t > 0 such that

r;

(X) c: e-tA(X) and e tAr * is a bounded

r operator from

Y

into X.

IV. There exists t> 0 such that rretA with domain X

t c: X is bounded as an operator from X into y.

o

V. There exists t > 0 and a continuous linear map

Q.

SX,A ~ Sy,S such that

r

= Q.e -tA. Proof.

I .. II. From the con tinui ty of r it immedia tely follows that

r

is con-r

tinuous. Its adjoint r*, defined by (x,r*y)X

=

(r x,y)y for all x € X, Y € V,

r r r

is a bounded operator as well. Now consider the dual operator

r' :

Ty,S ~ SX,A defined by

<F,r'G>x

=

<rF,G>y for all F € TX,A •

For fixed G € Ty S the right-hand side defines a continuous linear functional

*

'

on TX,A' rr is a restriction of

r'

and therefore maps

Y

into the set SX,A c: X.

*

II .. III. For each WEB the operator W{A)f_r :

Y

~ X is bounded and

1/1 (A>r; (y) c: Sx ,A' Similar to the me thod employed in Theorem 4.2 the resul t follows.

SA * 8.4.-* * S.4-* * I3A III ... IV. e 'T has a bounded adjoint (e '1') but (e 'T) ::>

r

e with

r r r r

domain Xs c X.

IV ... V. Take

Q

==

(eS~*)

*. For each a > 0 ;,Qa -aA is the extension of

r

r

e(S-a)A which is bounded. So 0 satisfies condition III of Theorem 4.2.

r -I3A ...

We have

r

== Qe •

-tA

V ... I. If

r ::

Qe the mapping

r

is obviously continuous. Theorem 4.6. Let ~ : TX,A -+ Ty,S be a linear mapping. Let 4>r denote the restriction of <P to X.

The following six conditions are equivalent.

C.S

I. 4> is con tinuous wi th respect to the strong topologies of T X A and Ty B'

_,

_,

II. For each g E SY,B the expression <g,4>F>y' F € TX,A' is a continuous

linear functional on TX,A'

I II. For eac h t > 0 the operator e -tB", . ... is a continuous map . f rom

T

X,A nto i SY,B'

IV. For each t > 0

-tB

*

For each t > 0 there exists 13 > 0 such that (e 4» (y)

SA -tB

*

r

e (e 4» is a bounded operator from

Y

into X.

r

V.

VI. For each t > 0 there exis ts 13 > 0 such that e -tB !fJ e I3A == e -tB",_BA '*""" on th e

r

domain X B C X is bounded as an opera tor from X in to Y.

Proof.

I .. II. Trivial.

-tB

I =11'111. Trivial, because e T_{y ,} B -+ Sy, B is continuous.

III .. IV. Theorem 4. 5, condi tion II. IV .. V. Apply Theorem 4.5 to e

-tB~

•

r

V .. VI. The adjoint of the bounded operator e SACe -tB!fJ ) * is an extension of

r

(e-tBt )e 13A. Therefore the latter is bounded.

r

o

VI =11'1. Let {Fn} be a nUll-sequence in TX,A' Then for each t > 0 there

exists 13 > 0 such that we can write (!fJF ) (t) == e-tBt e13AF (e). This converges

n -tB r 13A n to zero as n -+ 00 because of the bOUndedness of e t e •

c.,

II -v. <g,~F>V has the representation <f,F>X' see Theorem 3.2.IV, here f

=

~'g € SX,A' Taking F

=

u E SX,A we observe that (ul~'g)X

=

<u'~'g>X

=

=

<~/g>V is, as a function of g, a continuous linear functional on SV,B' Then with Theorem 4.2.VI it follows that ~. maps SY,B continuously into

SK A. Then, by Theorem 4.2. V for each t > 0 there exis ts

e

> 0 such that

el ....

-tB B t SA ... , -tB SA( -tB ... )* b f

e ~ e is a bOunded operator. u e ~ e

=

e e ~ ecause or r

all x E X, Y E Y

-tB -tB -tS -tB

(~'e y,xlx

=

<e y'¢rx>y

=

{y,e ~rx)V = «e ~r) y,x)X •

0

Theorem 4.7. Let the linear mappings

P

SX,A -+ Sy,B

r

TX,A -+ SV,B

~

TX,A -+ TV,B

K

SX,A -+ TV,B

be continuous with respect to the strong topologies on the mentioned spaces. Then the dual linear mappings

P'

r'

TY,B -+ TX,A Ty,S -+ SX,A SV,B -+ SX,A SV,S -+ TX,A

are also continuous with respect to the strong topologies. Proof.

Compare Theorem 4,2.v with Theorem 4.6.VI. compare Theorem 4,5.III with Theorem 4.S.IV. Look at Theorem 4.4.I1.

The interesting question arises which densely defined (possibly unbounded) operators ~rom X into V can be extended to a continuous mapping from TX,A into TV,S.

Theorem 4.8. Let E be a linear map X ::l D(E) -+ V with iJ('tf = X. E can be extended to a continuous linear map

E :

TX A -+ TV

B

iff

E

has a densely

*

*

"

defined adjoint E : V ::l D(E ) ::l SV,B -+ X with E*(SV,S) c SX,A'

C.7

Proof.

-... ) If E exists as a continuous map, its dual operator E' map~ Sy,B in to SX,A. For each x € D(E) and g € Sy

B

one has <g,Ex>y

=

(g,Ex)y

=

<E'g,x>X

=

=

(t·g,x)X. It follows that E*

~

E'

and E*(Sy,B) c SX,A •

• ) From Corollary 4.3 i t follows that E* maps Sy,S continuously into SX,A. Then by Theorem 4.7 the dual (E*)' maps TX,A continuously into TV,S.

However, (E*)' is an extension of E.

0

Corollary 4 _ 9 _ A continuous linear map Q. : Sx A .... Sv B can be extended to a

-

"

*

continuous linear map

Q :

TX A .... TV

B

iff

Q

has a Hilbert space adjoint

Q

*

* "

with D(Q) ~ SV,B and Q (SY,B) c

SX,A-CHAPTER 5. Topological tensor products of spaces of type SX,A' TX,A

For two separable Hilbert spaces

X

and

Y

we consider the complex vector space consisting of all Hilbert-Smidt operators Z from

X

into y. We shall denote this vector space by X ® Y _ For any Z € X ® V and any orthonormal

basis {e.} c X we have

~

co

III

Zlll2 =:

L

IIzei

I[~

< co _

i=l

The double norm

III ·111

does not depend on the choice of the orthonormal basis {e_i}. We in troduce an inner product in X ®

Y

by

(II)

Endowed with this inner product X ® V is a Hilbert space_ See [RS] Ch. VIII. 10. Examples of elements in X ® V are; ® n, ~ €

X,

n € Y, defined by

(; ® n)f

=

(f'~)Xn, for all f € X, and finite linear combinations of these:

E~=l (~j

® n_j) with

~j

€ X, nj € V. The linear subspace of X ® Y which

consists of all HS operators of the type just mentioned will be denoted by X ®a V, i.e. the (sesquilinear) algebraic tensor product of

X

and V.

X

® Y may be regarded as the completion of X ® Y with respect to the double norm

III-III. Therefore X till Y is called the completed (sesquilinear) topological tensor product of X and Y. For la ter reference we mention the following properties taken from [M

J

Ch. VIII.

Properties 5.1.

(a) Vx,~eX Vy,neY (;tilln,x@Y)XtiIIY = (x';)X(n,y)y • (b) VAeC V;eX VneY A(; @ n)

=

(~~) till n

= ;

@ (An)

C.8

Thus the canonical mapping

X

x Y +

X

@ Y, defined by [x;y] + x till Y is anti-linear in x and linear in y.

Let H respectively

1

denote bounded linear operators on ..

X,

respectively Y into themselves. H till

1

denotes the linear mapping of X till Y into

itself defined by

(H

@

1)

(x till y)

=

(Hx) @ (ly) and linear extension, followed by continuous extension.

(d) The uniform operator norms of HIland H @ 1 are related by

IIH till 111

=

IIHIIII1I1. (e)

(f)

_H

_and

₁

_{injective -}

_H

@

1

is injective.

The theory of closable tensor products of unbounded closable operators and the description of their properties in terms of corresponding properties of their factors presen ts greater difficulties. Only rather recen tly signif ican t results have been attained [T].

Definition 5.2. Let A with domain D(A} be a densely defined closed linear operator in X. Let B with domain D(B) be the same in Y. On DCA> till (B) c X till Y

a

we introduce the operator

A

till 1 +

I

till B by (A till 1 + 1 @ B) (x till y)

=

a a a a

=

(Ax) e y + x till (By) and linear extension. This extension is well defined

C.9

Lemma 5.3. Let A respectively B be self-adjoint operators in X respectively

Y.

I. A ~

1

+

1

~ B is essentially self-adjoint in X ~ Y. We denote the

a a

unique self-adjoint extension by A ~

1

+

1

~ B or, briefly, A E B. I I . A ~ 0 and B 2': 0 implies A E B ~

o.

Proof. As in [W] section B.S.

Theorem 5.4. On X ~

Y

we have for t ~

°

-t(AEB) -tA -tB

e · =e ~e •

Proof. As in [W] section B.S.

Applying the results of the preceding chapters we can introduce the spaces SxeY,AEB' TX~y,AEB and, by taking A

=

0 or B

=

0, the spaces SX~y,AEl' T

X~Y ,IEB'

etc.

Definition 5.5. The canonical sesquilinear map ~ : SX,A x Sy,B ~ SX~y,AEB is defined by [u;v] ~ u ~ v. Here the symbol ~ is the same as in Properties 5.1. This definition is consistent because for u e Sx A' v e Sy B there

-tA ' -tA '

exist x e X, y e Y and t > 0 such that u

=

e x, v

=

e y. Further,

-tA -tB - t A - t B

u ~ v

=

(e ~ e ) (x ~ y)

=

(e x) ~ (e y), so that u ~ v e SXeY,AEB'

Theorem 5.6. SXey,AEB is a complete topological tensor product of SX,A and Sy,B' By this we mean:

I. SX~Y ,AEB is complete,

II. The canonical se~quilinear map ~ : SX,A x Sy,B + SX~y,AEB is continuous. III. The span of the image of ~t i.e. the algebraic tensor product

SX,A ~a Sy,B' is dense in SX~y,AEB' Proof.

I. The completeness follows from Theorem 1.11.

I I . It is enough to check the continuity of e at [0;0]. Let W be a convex open neighbourhood of 0 in SX~y ,AEB' Then for each t > 0, W n (X ~ y) t

C.10

is an open a-neighbourhood in (X @ Y> t and i t con tains an open ball centered at 0 and radius rt' 0 < r

t < 1. In Xt respectively Yt we introduce open balls At respectively Btl centered at a and with radius both r

t• Let

and

Then @ maps A x B in W since

B

=

U _Bt c SY,B •

t>O

whenever x € A,

Y

€ B. Let

A

respectively

B

denote the convex hulls of A respectively B. Then @ maps

A

x

B

in W. The set

A

is convex and A

n

X

t contains an open neighbourhood in Xt• From Theorem 1.4.II follows that

A

contains an open set U,I. •

""e:

Similarly

B

contains an open set V ~. We conclude that @ maps

X,\)

U x V ~ into W.

1j!,e: X,\)

i t

III. For each t > 0, X

t @a

Y

t is dense in (X @ YI t' From this the desired resul t follows.

Remark. Our strong topology on SX@V

,AatB

is, generally speaking I not the projective tensor product topology. Cf. [SCH] p. 93. Therefore the universal factorization property for continuous sesquilinear maps on this space does not hold in general.

Definition 5.7. The canonical sesquilinear map @ : TX,A x TY,B ~ TX@V,AatB' [FiG] ~ F @ G, is defined by (F @ G} (tl

=

F(tl @ G(t),

o

Here @ is the same as in Properties 5.1. The definition is consistent because (F @ G) (t + T)

=

e-·A F(t) @

e-·

B

G(t)

=

=

(e-TA @ e-TB]F(tI @ G(t))

=

(e-TA @ e-TB) (F @ G) (t) •

Theorem 5.8. T X@V,AsB is a complete topological tensor product of T X,A and TV,B' By this we mean:

C.ll

II. The canonical sesquilinear map @ : TX~A x TY,B ~ TX@Y,AmB is continuous. III. The span of the image of @, i.e. the algebraic tensor product

TX,A @aTy,B' is dense in TX@Y,AmB" Proof.

I. The completeness follows from Theorem 2.5.

II. For each t > 0 we have ([F (t) @ G (t) [X®Y

=

[F (t) [X[G (t) II Y" From this the continuity at [O;oJ follows.

III. X ®a Y is dense in X ® Y which is dense in TX®Y,AmB.

Now mixed sesquilinear topological tensor products of type SX,A @ Ty,B' TX,A ® SY,B will be considered. The notation of [EThJ, Ch. II, will be used.

Definition 5.9. We introduce the following linear subspace of T_X®Y,I®B!

In this space we take the topology generated by the semi-norms

Definition 5.10. The canonical sesquilinear map

is defined by

f ® G : t ~ f II G{t) •

(It is clear that Vt>O f ® G(t) € SXIIY,A®I.1

Theorem 5.11. T(SX®Y,A@I,1 ® B) is a complete topological tensor product of SX,A and Ty,B' By this I mean

I. T(SX®Y,A®I,1 ® B) is complete. II. The canonical sesquilinear mapping

is con tinuous.

C.12

Proof

I. Let {<Pal be a Cauchy net. The ats belong to a directed set D. First take TjJ == 1. (l a tends to a limi t poin t <P EO T X€OY , 1 €OB because the latter

is complete. It remains to show that, for each t > 0, iP(t) € SX€OY,AeI"

For each TjJ EO B' and each t > 0, (!/I (A) & 1)41 == TjJ(A &

niP

converges in

+ a a

X & Y. From the closedness of TjJ(A &

1)

it follows that ~(t) € D(TjJ(A & 1».

This is true for each TjJ € B+ and therefore by Theorem 1.10, for each

t > 0, <P(t) € SX€OY,A€OI'

II. Let TjJ € B+ and let t > O. Then for f € SX,A and G € TY,B

From this inequali ty the con tinui ty of & follows.

III. Since SX€OY,AmB is dense in _{T(SX€OY,A€OI,1 e B) and since SX,A ea SY,B is}

dense in SXeY,AmB' the assertion follows.

0

Definition 5.12. We introduce the following linear subspace of T_XeY,Ael:

In this space we take the topology generated by the semi-norms t > 0 , ql € B+ •

Definition 5.13. The canonical sesquilinear map

is defined by

F & g : t ~ F(t) & g •

(It is clear that Vt>O F(t) & g EO SXeY,leBo)

c.13

Theorem 5.14. T<SxeY,leB.A e 1) is a complete topological tensor product of T

X,A

and Sy,

B"

Hence

I. T($XeY,leB,A e

1)

is complete. II. The canonical sesquilinear mapping

is con tinuous.

Next I introduce a second type of mixed topological tensor product.

Definition 5.15. We introduce the following linear subspace of TXeY,AeI: -tB

T

$ <TXeY ,Ae1'

1

e B)

=

u

(Z e e ) ( XeY AeI) •

t>O '

-tB

Each of the spaces (1 e e ) (TXeY,AeZ> can be written as TXee-tB(Y),AeZ' According to Chapter II they are Frechet spaces, Their semi-norms are given by

n E IN,

The space S(TXeY ,AeI,1 e B) is an inductive limit of the spaces

(1 e e-tB) (TXeY,Ael)' For its topology we take the inductive limit topology.

On _{TCS XeY ,Ae1,1 e B)} x $(TXeY,AeI'Z e B) we introduce the pairing

for € > 0 sufficiently small. Finally, we define the embedding of S<TXeY ,Ael,I e B) into T(SXeY,leB,A e I} by

C.14

Theorem 5. 16.

I.

TX,A

~a

SY,B

is dense in S(TX~y,A@I,1 ~

B).

II. For each fixed ~ €

T(SX@Y,A01,1 @ B)

the linear functional

F

+ <~,F>B

is continuous on S(TX~y,A@111

@

B).

III. The embedding described in Definitions 5.15 is continuous. Proof.

I.

SX,A

~a

SY,B

is dense in

SX®Y,AaB'

which is dense in S(TX~y,A~111 ~

B).

II. It is sufficient to prove the continuity for !!estrictions to each space

-tB

(1 ®

e )

(T

_X

®Y,A®I)'

We have

From this the continuity follows.

III. Let t > O. Let

W

€ B+. Let a > 0 be fixed. Then for W € TX®y,A~1

-aB

A

So emb 0

(1

& e ) is continuous from

TX®Y,A®1

into T(SX~y,1®B' ~

1).

From this the con tinui ty of emb follows.

0

Defini tion 5. 17. Similar to Defini tion 5.15 we in troduce the space

wi th the appropriate inductive limi t topology.

C.lS

Analogous to Theorem 5.16 we have Theorem 5.18.

I.

SX,A e

_a

Ty,B

is dense in

S<TXeY ,Ael,1 e

B).

II. For each fixed

P

€

T(SXeY,Ael,1 e B)

the linear functional T +

<P,r>A

is con tinuous on

S <T XeY ,Ae l' 1 e

B)"

III. The embedding described in Definition 5.17 is continuous.

Remark. For more details on the topological properties of the spaces

T(SxeY,Ael,1 e B)

,ete. I see [ETh].

CHAPTER 6. Kerneltheorems

In this final chapter we show that the elements of the completed sesquilinear topological tensor products of the preceding chapter can, in a very natural way, be interpreted as linear maps of the types we discussed in Chapter 4.

-tA

-tB

We give necessary and sufficien t condi tions on the semi -groups e and e which ensure that the topological tensor product comprise all continuous linear maps. In this case we say that a kernel theorem holds.

CASE a: Continuous linear maps

TX,A

+

SY,B"

We consider an element

e

€

SxeY,AEB

as a linear operator

TX,A

+

Sy,B

in the following way.

Let F €

TX,A"

We define SF by

For e: > 0 and sufficien tly small this definition makes sense and does not depend on e:.

Theorem 6.1.

I. _{For each S} €

SXeY,AEB

the linear opera tor S

(al, is continuous.

C.16

II.

-tA -tB

III, If for each t,. 0 at least one of the operators e , e , is

as,

then SXsy,AsB comprises all continuous linear operators from TX,A into SY,B. IV. SXSX,AeA comprises all continuous linear operators from TX,A into SX,A

-tA

iff for each t > 0 the operator e is

as.

Proof.

I. We shall prove that e satisfies condition IV of Theorem 4.5. Since e € X S Y we have S = S. Since e € SX4Y A B we have for t sufficiently

tB

tA

r .... ,IJ tA

small e Se If: X S Y. Therefore,

aetA

=

e-tB(e tB Se ) is bounded.

II. For t sufficiently small etB SetA If: X S Y, therefore by Properties

5.1. c

<SF,G>y

=

<e -tB tB (e ee )F(t),G>y tA

=

«e

e:8 Se tA )F(e:) ,G(e;)y

=

tB tA

=

(e ee ,F(t) S G(t»XsY

=

<e,F S G>XsY •

III. Let

r :

TX A + Sy B be continuous. By Theorem 4.5.V there exists T > 0

,

,

-~

and continuous

Q :

Sx A + Sy B such that

r

=

Qe • By Theorem 4.2.v

,

'aB

-~TA

there exis ts

a

> 0 such tha t e

Q

e is a bounded opera tor, Pu t

a - ~ min(a,~T), then

The opera tor be tween ( 1 is bounded, Further the opera tor be tween { } is

as

since e-(S-alB or

e-(~T-alA

is

as.

It follows that

r

IS SXsY,AIJB'

IV. The if-partis a special case of III. For the only-if-part consider the -aA T

special map

r

=

e : X ,A + Sx ,A for some a > O. In order that

C.l?

CASE b: Continuous linear maps SX,A + Ty,S"

Let K € TX@Y,AaS- For f € SX,A we define Kf € Ty,S by

For any f € SX,A and t > 0 this defini tion makes sense for e: > 0 sufficiently small_ Moreover, (Kf) (t) does not depend on e:.

Theorem 6.2.

I. For each K € T~y,AmS the linear operator K SXtA + Ty,S defined by

(b) is continuous.

II. For each K € TX@y,AaS' f € SX,A' g € Sy,S'

III. If for each t > 0 at least one of the opera tors e

-tA

, e -tB is ElS, then . TX@y,AIBS comprises all continuous linear operators from SX,A into Ty,S" IV. _{TX@X,AaA comprises all continuous linear operators from SX,A into TX,A}

-tA

iff for each t > 0 the operator e is ElS. Proof.

I. We use condition II of Theorem 4.4.

-tB K -aA

For each t > 0, a > 0, e e is a bounded opera tor from X in to . Y because for e: sufficiently small

-t8 K -etA -(t-e:)S K( ) -(a-e:)A

e e

=

e e: e •

All operators in the last expression are bounded.

II. For each T > 0, K(T) is a as-map. For e: sufficiently small, with Properties S.l.c

K e:S -e:S -e:A e:A

C.1S

III. Let

L :

Sx A

~

Ty B

be continuous. According to Theorem 4.4.11 the

~~ ~U

-d

operators e Le are bounded for each t > O. However, if e or

-tA

-tB

-tA

e is HS for each t > 0, then e Le is HS for each t >

a

and it defines an element in T

X&Y ,AIIIB"

A simple verification shows that this element reproduces L by recipe (b).

IV. The if-part is a special case of III. For the only-if-part consider the special map L = emb =

1.

Here

1

is the iden ti ty map.

-tA 1 -tA -2 tA b . d 1 f S ~ff

e e

=

e can e cons~dere as an e ement 0

X&Y,AsB.

e -tA is

as

for all t >

o.

D

CASE c: COntinuous linear maps:

SX,A

~

SY,B'

Let P €

T(SX&Y,r®B,A & 1).

For f €

SX,A

we define pf €

SY,B

by

(c) Pf

=

Pee) e eAf •

Pf€

SY,B

since Pee) €

SX®Y,I&B'

The definition makes sense for €

sufficient-ly small and does not depend on the choice of e.

Theorem 6.3.

I. For each P €

T(SX®Y,I®B,A & I)

the linear operator P

SX,A

~

Sy,B

defined by (cl is continuous.

II. For each P E

T(SX&y,I®B,A ® 1),

each f E

SX,A'

each G E

Ty,B'

III.

IV.

-tA -tB.

If for each t > Oat leas t one of the opera tors e , e ~s

as,

then

T(SX&Y,I&B,A & I)

comprises all continuous linear operators from

SX,A

into

SY,B'

Consider the special case

X

=

Y

and

B

=

A.

The space

T(SX&X,I&A,A & I)

comprises all continuous =!Aear operators from

SX,A

into itself iff each t >

a

the opera tor e is HS.

Proof.

I. We use condi tion II of Theorem 4.2. Let f

n -+- 0 strongly in Sx A' For some e: > 0, e sAf -+- 0 in X.

C.19

, n 158

SX0V 10B' therefore there exists <5 > 0 such that e P(s) is a

, 5B oB e:A

Pes) IS

bounded operator. But then e Pf == (e P(s»e f -+- 0 in V, which

n n

shows that Pf

n -+- 0 strongly in SV,B'

II. For B and <5 sufficien tly small and posi ti ve we have

III. Let Q. : Sx A -+- Sv 8 be continuous. According to Theorem 4.2.V for each

" B(t)8 -tA

t > 0 there exis t B ( t) > 0 such that e Qe is a bounded map from X in to V" NOW because of the assumption on

e-a~

e -0.8, we find that Q.e-tA

=

e-S(t)B(eB(t)B Q.e-tA) is an element of T(SX&V,1&B,A @ 1). It reproduces the operator

Q.

if the recipe (c) is applied.

IV. The if-part is a special case of III. For the only-if-part consider the

-tA

identity map 1 : SX)A -+- SX,A" In order that Ie

tA

as a function of t is an element of T(SX&X,I&A,A @

11

the operator e- has to be

as

for all

t > O.

CASE d: Continuous linear maps: T X,A -+- TV ,8"

Let ~ € T(SX@V,A@I,r 0 81. For F € TX,A we define ~F E TV,B by

(d)

(~F)

(tl

=

w(tles(tlA F(e:(t» •

This definition makes sense for t > 0 and e:(t) > 0 sufficiently small. ($F) (~ € SV,8 since ~(t) € SX@V,A@1" Moreover

o

C.20

Theorem 6.4.

I. For each t E _{TCS X0Y ,A01,1 0 B) the linear operator} ~ TX,A + Ty,B defined by (d) is con tinuous.

III.

IV.

-tA

-t8

If for each t > 0 at least one of the operators e I e is

as,

then TCS

X0Y,A01,1 0 B) comprises all continuous linear operators from TX,A into TY,B'

Consider the special case Y

=

X and B

=

A. The space T(SX0X,A01,1 0 A) comprises all continuous linear operators from TX,A into itself iff

-tA

for each t > 0 the opera tor e .is

as.

Proof.

I. We use Theorem 4.6.1II. For each t > 0, e

-tB

~ E SX0Y,A01. Then according

-t8

to case a, e ~ is a con tinuous linear map from T X itA in to Sy, B' II. For a and 0 sufficiently small and post tive

aB

~,F 0 ~B

=

<~Ca),(1 0 e ) (F 0 g}>X0Y

=

aB oB oA (a-olB )

=

<~«l) IF e e g>XeY = (e ~(a)"e ,F(o). e 9 X®Y =

oA aB

=

(~(a}e F(ol,e g)y

=

III. Let V : TX A + Ty B be continuous. According to Theorem 4.6.VI for each

"

-tB

6

(t)

A

t > 0 there exists I) (t) >

°

such that e V e is a densely

r -aA

defined and bounded operaror from X into

Y.

If one of the operators e

~B.

-tB

e l.S

as

for arbitrary small positive a i t follows that e V is

as

r

for t > 0, because

..

C.21

IV.

Here 'l' e S (t) A denotes the extension of 'l' e a (t) A to the whole of X.

r r Since -tB e 'l' r i t follows belongs to duced. -~tB ( -~tB \II B (~t) A) -B (~t) A = e e T e e r

-tB

-tB

that e 'l'r €

SXeY,A®I-

Hence e 'l'r as a function of t,

T(SX®Y,Ael,1 ®

B). By recipe (d) the operator 'l' is

repro-The if-part is a special iden ti ty map

I.

In order sidered as an elemen t in HS for all t > O.

case of III. For the only-if-part consider the

-tA

that e ,as a function of t, can be

con--tA

T(SX®X,A®T,r ® A)

the operator e should be

Remark. In [ETh] a kernel theorem for extendable continuous linear mappings has been stated and proved.

REFERENCES

See part A.

ACKNOWLEDGEMENT

The author thanks Dr. S.J.L. van Eyndhoven for contributing to the revision of the manuscript.