An operator variant of the maximum principle - Springer Link

4.

B. F. By!ov, R. E. Vinograd, D. M. Grobman, and V. V. Nemytskii, ponents and Its Applications to Stability Problems [in Russian], V. M. Miilionshchikov, "Baire classes of functions and Lyapunov ents. Uravn., 18, No. 5, 804-821 (1982). V. M. Millionshchikov, "Baire classes of functions and Lyapunov ents. Uravn., 18, No. 6, 957-978 (1982). V. M. Millionshchikov, "Baire classes of functions and Lyapunov ferents. Uravn., 1-8, No. 8, 1330-1345 (1982). V. M. Millionshchikov, "Baire classes of functions and Lyapunov ents. Uravn., 1-8, No. 9, 1507-1548 (1982). V. M. Millionshchikov, "On the typical properties of conditional II," Differents. Uravn., 19, No. 9, 1503-1510 (1983). V. M. Millionshchikov, "On the typical properties of conditional IV," Differents, Uravn., 20, No. 2, 241-257 (1984).

5. 6. 7. 8. 9. I0.

Theory of Lyapunov ExNauka, Moscow (1966). exponents. VI," Differexponents.

VII, ~' Differ-

exponents.

VIII~" Dif-

exponents.

IX," Differ-

exponential stability. exponential stability.

AN OPERATOR VARIANT OF THE MAXIML~ PRINCIPLE V. G. Kurbatov and I. S. Frolov

In [I] one has given an analogue of the maximum principle for the case of an analytic function~ taking values in a Banach algebra. In the present note, this result is refined and carried over to a more general situation. All the linear spaces will be assumed to be complex and all the algebras contain an idenThe spectrum of an element x of an algebra B will be denoted by o(x) = OB(X).

tity.

Let ~ be a fixed Banach algebra, and let X be a compact topological space. By C (X, ~) we shall denote the Banach algebra of all continuous functions x: X - + ~ with pointwise multiplication and with the norm llxl[= m a x ]Ix (t)II. We note that for x ~ C (X,~) one has the formula

': (~) = U (': (x (t)): t ~ x } . Definition (see [2, pp. 95, 38]). Let a ~ C be a compact set. By the polynomial convex hull we mean the set po a ~ G , the union of o and all the bounded connected components of the complement C \ a . THEOREM i.

C=

Let F be a closed subset of X and let A be a closed subalgebra of the algebra

C (X,~) , possessing the property: A)

for any

Then for any

x~A, x~A

the function t~+ Hx (t) I] attains its maximal value on F. we have (JA (x) C po ~ (x, F),

where

a (x, r) = U {(J (x (t)): t ~ r}. We note that condition A) is usually satisfied in the following stronger form: A') for any x ~ N and I ~ $ * (here s is the conjugate of ~ , viewed as a Banach space), the scalar function t ~ ll(x (t)) I attains its maximal value on F. The proof of Theorem 1 will be based on the following statement. Proposition 1 [2]. we have

Let A be a closed subalgebra of the Banach algebra B.

Then for any

x.~_A

~

(x) C po ~B (x).

Kuibyshev State University. Translated from Matematicheskie Zametki, Vol. 36, No. 4, pp. 531-535, October, 1984. Original article submitted April 28, 1983.

0001-4346/84/3634-0763508.50

9 1985 Plenum Publishing Corporation

763

The Proof of Theorem i. We denote by B the algebra C (F, Z) . canonically in B. It remains to refer to Proposition I.

The algebra A is imbedded

Example i. Let X be the circle ~ -= { ~ E : I~ ] :~J i}, A = A (U, ~) is a subalgebra of functions x ~ C (~7, fs , analytic inside U. Here, obviously, condition A') holds for F = 3U (by ~U we mean the topological boundary of the set U). Example 2. Let ~ ..... ~ , ~ C C be compact sets, fl = QI x . . . x fl~, A = A (f~,f~) is a sub, algebra of functions x ~ C = C (fl,~'Z:), analytic with respect to each variable z~.~ ~ (k ----I, ..., n) inside ~k, the remaining variables being fixed. Here condition A') holds for Y = 0 ~ X ... X 0 ~ 9 Theorem i leads to the following estimate:

~A (X) ~ po a (x, F) We mention the obvious equality ~A(X) -- oC(X).

(x ~ A).

In [i] one has proved the weaker assertion:

~c (x) C co o (x, r) ( h e r e co d e n o t e s

the operation

of taking

the convex hull).

We p r o c e e d t o d e s c r i b e t h e m o s t i n t e r e s t i n g c o n s e q u e n c e o f Theorem 1. L e t M b e a commut a t i v e Banach a l g e b r a , l e t X b e t h e s p a c e o f c h a r a c t e r s o f M, l e t ~: M - ~ C (X, C) b e t h e G e l f a n d t r a n s f o r m , and l e t F C X b e t h e S h i l o v b o u n d a r y ( s e e [3, Sec. l l ] , [4, Sec. 1 3 ] ) o f t h e a l g e b r a M. We denote by M norm [5]. We shall l[x~x~ll ~ llx~l]'IIx~]I est cross-norm [5]) with the algebra C

~ f~ the tensor product M ~ 7', completed with respect to some crossassume that the cross-norm a is consistent with multiplication [6], i.e., for any x~, x ~ 2 ~ l @ Z . The algebra C ( X , E ) ~ e ~ g (where r is the smallis identified by virtue of the canonical isomorphism [7, Chap. 2, p. 89] (X, f~).

We consider the mapping I

THEOREM 2.

For any x ~ M ( ~ Z

we have

(x) C po a ((I)x, F), where

((l)x, r) = U {~ I(@x) (t)]: 1 ~ Y}. The proof of Theorem 2 will be based on the following statement, results of [8-10]. Proposition 2.

For any

easily derived from the

z ~ M @ r~fg we have o (x) =

o c (x,~) ( @ x ) .

Proof of Theorem 2. We denote by A C C (X, fg) the closure of the image Im~ of the mapping r Obviously, the algebra A has property A'). Applying Theorem i to the element ~bz ~ A, we obtain that ~a (~Dx) C po a (qbx, F). mate

In order to conclude the proof, it remains to apply Proposition 2 and the obvious estia~(x,z ) ((I)x) C oa ((:I)x). We note that Examples I and 2 are covered trivially by Theorem 2. C) ~)~fffg,

and,. therefore,

For example, A(U, ~') =

~ is the identity mapping.

Example 3. We denote by H ~ (U, fg) (where U = { k ~ C : I~ I < I} ) the Banach algebra all bounded analytic functions x: U - + Z , whose range x(U) is relatively compact in ~ ,

of with

the norm IIx II= sup {Ifx (t) If: t ~ U} . In the case of a finite-dlmensional algebra ~ , the requirement of relative compactness does not impose any restriction. It is needed in order to have the easily verifiable isomorphism H ~ (V, C) (>~ X :~_ H ~ (U, X )

The algebra H ~ (U, C) is commutative and, consequently, the conditions of Theorem 2 hold. The space of the characters of X and the Shilov boundary F of the algebra II~ (U, (i) are de-

764

scribed, for example, in [ii, Chap. i0]. result.

The application of Theorem 2 leads to the following

Let x~_7/~ (U, ~) . Then for a.a. z ~ S U the functions xr (z) ~ z (rz) (r~ (0, i)) converge for r + i to some function 2 ~ L ~ (~U, ~ [Ii]. We have the inclusion

Example 4. Let E be a Banach space, let ?i: be the algebra of all linear bounded operators in E. We denote by C(R, E) the space of all continuous bounded functions x: R-~/r with the

norm

l]xi[

: s . p il x ( / ) ! i ii

Let a~ ::~ ~, /q

9

~)(k~ N), and _~ tVi" H a~ Jill< oo .

acts in C(R, E) and it is bounded. V(~;) 9

Relative to the norm

Then the operator

The set of such operators forms an algebra, denoted by

ilD Ji:~ ~ IIa~]I it is a Banach algebra.

The investigation of the

stability of functional-differential equations leads [12] to the study of invertibility in the algebra V (iS 9 Clearly, V(C) is commutative and

V (C)(•

(%) (where ~ is the largest cross-norm

[5, Sec. 3]). Without dwelling on technical details, we give an estimate obtained in the case under consideration with the aid of Theorem 2:

a~-(~)(D)Cpo ~ (D, r), where o(D, r) is the closure of the set

We note that in this example ~D is realized as an analytic function of infinitely many variables on U~. We also emphasize that the set o(D, F) coincides indeed [8] with the usual spectrum of the operator D. Therefore, the described estimate can be obtained also from Proposition io LITERATURE CITED I. 2. 3.

V. Eo Slyusarchuk, "Estimates of spectra and the invertibility of functional operators," Mat. Sb., 105, No. 2, 269-285 (1978). N. Burbaki (N. Bourbaki), Spectral Theory [Russian translation], Mir, Moscow (1972). I . M . Gel'fand, D. A. Raikov, and G. E. Shilov, Commutative Rings, Chelsea, New York

(1964). 4. 5. 6. 7, 8. 9. i0.

!i. 12.

M. A. Naimark, Normed Rings [in Russian], Nauka, Moscow (1968). R. Schatten, A Theory of Cross-Spaces, Princeton Univ. Press (1950). B. Simon, "Uniform crossnorms," Pac. J. Math., 46, No. 2, 555-560 (1973). A. Grothendieck, "Produits Tensoriels Topologiques et Espaces Nucleaires," Mem. Lm. Math. Soc., No. 16 (1955). S. Bochner and R. S. Phillips, "Absolutely convergent Fourier expansions for noncommutative normed rings," Ann. Math., 43, No. 3, 409-418 (1942). G. R. Allan, "Ideals of vector-valued functions," Proc. London Math. Soc., 18, No. 2, 193-216 (1968). I. Gokhberg and Yu. Laiterer (J. Leiterer), "The factorization of operator-functions relative to a contour. II. The canonical factorization of operator-functions that are close to the identity operator," Math. Nachr., 54, No. 1-6, 41-74 (1972). K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs (1962). V. G. Kurbatov, "On the solvability with respect to the derivative of a stable functionaldifferential equation," Ukr. Mat. Zh., 34, No. i, 103-106 (1982).

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