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We construct a Sobolev-type space of almost periodic functions, in which we study differential and pseudodifferential operators. We show that the usual theorem on the regularity of solutions of elliptic equations is not true in these spaces, and we present modified variants of this theorem. We also study the question of the defect indices of differential operators in the space of Besicovitch almost periodic functions, and prove a Liouville-type theorem in this space. Bibliography: 16 items.

The notation of Gramian matrix is generalized to operators between function spaces and then used to provide an operator-valued inner product between pairs of Hilbert space-valued Banach function spaces. In this context projection operators, Schwarz's inequality, and orthonormal series are discussed.

Let H be a fixed Hilbert space and B(H, H) be the Banach space of bounded linear operators from H to H with the uniform operator topology. Oscillation criteria are obtained for the operator differential equationwhere the coefficients A, C are linear operators from B(H, H) to B(H, H), for each t ≤ 0. A solution Y: R+ → B(H, H) is said to be oscillatory if there exists a sequence of points ti ∈ R+, so that ti → ∞ as i → ∞, and Y(ti) fails to have a bounded inverse. The main theorem states that a solution Y is oscillatory if an associated scalar differential equation is oscillatory.

The spectral mapping theorem for joint approximate point spectrum is proved when A is an «-tuple of commuting operators on a Banach space and ƒ is any w-tuple of rational functions for which f (A) is defined. The purpose of the paper is to show how properties of parts of the joint spectrum can be obtained by use of spaces of sequences of vectors. The method (first used for general Banach spaces in [4]) provides very simple proofs of some known results ; but the main result is believed to be new. Throughout the paper, we deal with n-tuples of bounded operators on a Banach space 2£ 9 whose vectors are denoted by x, y, • • •. It will be convenient to use a symbol such as A to denote an «-tuple of operators and a symbol such as A to denote an «-tuple of complex numbers: A = (Al9 • • • , An)9 A=(Al5 • • • , AJ, 0=(0 , • • • , 0). The one-sided spectra of interest in this paper are not defined in terms of existence of Banachalgebra inverses, the context of the papers of Robin Harte ([5], [6]); they are the somewhat different spectra of the following definition. (However, in the special case that 2£ is a Hilbert space, our result is a corollary of Harte's.) DEFINITION. We say 0eov(A) in case there exists a nonzero xe& such that Ax=0 (i.e., AjX=0 for j=l, • • • , « ) . We say 0 e av(A) in case for every £>0 there exists a unit vector xeSC such that \Ax\

We first present an elementary commutation theorem for operator algebras on Hilbert space. This theorem is then applied to give an elementary proof of duality for free Bose fields.

The concept of semi-normality and hyponormality are defined for unbounded operators in Hilbert Space and Commutators AB-BA, with A unbounded and seminormal and with B bounded, are studied. Four theorems are proved. Two of them are generalizations of results of C. R. Putman.

Introduction. The question of ergodicity of a semigroup of bounded linear operators on a Banach space has been reduced, by Alaoglu and Birkhoff [1],' Day [2, 3], and Eberlein [4], to the study firstly of the ergodicity of the semigroup itself and secondly, of the ergodicity of each element of the Banach space with respect to this ergodic semigroup. In the case of a bounded and commutative semigroup operating on a reflexive Banach space, the ergodicity has been established [2, Corollary 6], [4, Corollary 5.1, Theorem 5.5]. For noncommutative semigroups, Alaoglu and Birkhoff have proved ergodicity by restricting the Banach space [1, Theorem 6]. A general discussion of the noncommutative case is given by Day [3 ]. The following is an attempt to prove ergodicity, in the strong topology, for a certain noncommutative case, where we use a weakened form of the commutative law. Our method uses a fixed point theorem of Kakutani [5], and a theorem about the group of cluster points of a directed subsemigroup, which we have developed elsewhere [6]. It should be observed, however, that the theorem which is proved here, though being similar to, is not a generalization of the results mentioned above. In fact it is apparently independent of them, because we postulate here a compactness in the strong topology of operators; whereas in the analogous theorems of Day and Eberlein, a kind of weak compactness is used, by considering bounded semigroups acting on reflexive spaces, a natural legacy from the study of Hilbert space. However our theoremn is a generalization of a theorem of Yosida [7, Theorem 2 ].

Problems concerning equilibrium strategies for linear games are treated in a Hilbert space setting. First a nonlinear equation is derived for the equilibrium solutions in the player’s allowed operator feedback spaces. The results obtained from the study of this equation are then used to compute the solution to a class of differential games and to establish the uniqueness of the solution.

A representation of functions as integrals of a kernel ψ ( t ; x ) \psi (t;x) , which was introduced by Studden, with respect to functions of bounded variation in [ 0 , ∞ ) [0,\infty ) is obtained whenever the functions satisfy some conditions involving the differential operators ( d / d t ) { f ( t ) / w i ( t ) } , i = 0 , 1 , 2 , … (d/dt)\{ f(t)/{w_i}(t)\} ,i = 0,1,2, \ldots . The results are related to the concepts of generalized completely monotonic functions and generalized absolutely monotonic functions in ( 0 , ∞ ) (0,\infty ) . Some approximation operators for the approximation of continuous functions in [ 0 , ∞ ) [0,\infty ) arise naturally and are introduced; some sequence-to-function summability methods are also introduced.

The trace of a positive bounded operator in a Hilbert space is defined and shown to be independent of the basis used in the definition. The trace of the product WA is defined for bounded A and positive bounded W with unit trace and is also shown to be basis‐invariant. An integral representation is derived and used to define Tr(WA) for unbounded A. Linearity, isotony, and continuity with respect to uniform covergence are proved for Tr(WA) as a function of (bounded) A. Several formulas used in statistical mechanics are derived, and it is proved that if A and B are commuting positive operators, then Tr(W(A + B)) = Tr(WA) + Tr(WB). A counter example is given which shows that the commonly used definition of Tr(WA) in terms of an orthonormal basis is not invariant under permutation of the basis vectors even in the case of a very simple unbounded operator A.An expectation‐value function M which associates the expectation value M(A) to the operator A is assumed given and subject to certain restrictions. For bound...

Introduction. In a recent paper [17] we obtained a spectral representation for the bounded normal operator solutions of a certain functional equation whose special cases include the semigroup equation (U,,, = U,LUV) as well as many others. In this note we continue our study of the abstract functional equation expressed in terms of algebras of measures. We shall consider therefore a mapping z -* UZ of a locally compact space Z into the set of unbounded normal operators on a Hilbert space. Since the operators are not necessarily bounded, the conditions imposed on the mapping z -* Uz vary somewhat from the conditions in the bounded case. The main result (Theorem 1) which we obtain here is a spectral representation analogous to the result valid in the bounded case [17; 18, Theorem 2, p. 8]. The methods which we use are suggested by the papers [12] and [13] of C. Ionescu Tulcea. For the general theory and terminology of unbounded operators we refer the reader to the works of E. Hille and R. Phillips [10], C. Ionescu Tulcea [11], F. Riesz and B. Sz.-Nagy [21]. Spectral representations for semigroups of unbounded operators are also discussed in A. Devinatz [3], A. Devinatz and A. E. Nussbaum [4; 5], R. Getoor [7], C. Ionescu Tulcea and A. Simon [14] and A. E. Nussbaum [19]. Theorem 1 of this paper contains the main part of Theorem 1 in [14] (and henceforth certain of the results given in [3; 4; 5; 7; 19]). The setting of the problem in algebras of measures is given in ??1, 2, and 5. Spectral families of Radon measures and direct sums thereof are summarized in ??3 and 4. The main result is stated and proved in ?6. In ?8 we apply the main theorem to derive a corollary which in the bounded case extends a result of S. Kurepa [16].

Recall that the spectrum, σ(T), of a linear operator T in a complex Banach space is the set of complex numbers λ such that T—λI does not have a densely defined bounded inverse. It is known [7, § 5.1] that σ(T) is a closed subset of the complex plane C. If T is not bounded, σ(T) may be empty or the whole of C. If σ(T) ≠ C and T is closed the spectral mapping theorem, is valid for complex polynomials p(z) [7, §5.7]. Also, if T is closed and λ ∉ σ(T), (T–λI)−1 is everywhere defined.

One of the most useful tools for dealing with the theory of nonlinear operators in concrete analytical problems is the Leray-Schauder fixed point theorem [5]. A disadvantage is that it is only available for the completely continuous (that is, compact and continuous) operators, and moreover the proof of the theorem uses the theory of the topological degree of a mapping. However, in a recent publication, Browder [1] has given an elementary proof without recourse to degree theory, tt was observed by the authors that this method of proof is applicable to a wider class of operators, namely those we shall call strongly P-compact. In this note we shall show that for a restricted class of Banach spaces (including, however, all separable Hilbert spaces) the class of strongly P-compact operators is quite extensive, and shall prove a fixed point theorem like that of Leray-Schauder for this class. Throughout this paper X will denote a real Banach space which has the following property: (re)l: There exists a sequence {F.} of finite dimensional subspaces of X, with F.CF,+I for each n, and U F, dense in X, and a corresponding sequence

We prove that a normal operator on a separable Hilbert space can be written as a diagonal operator plus a compact operator. If, in addition, the spectrum lies in a rectifiable curve we show that the compact operator can be made Hilbert-Schmidt.

The purpose of this paper is to extend a well-known theorem of Lyapunov on matrices to unbounded linear operators in an infinite-dimensional Banach space by using semigroup theory. This extension is used for the study of stability problems of nonlinear differential equations. In the special case of a Hilbert space, a direct extension of Lyapunov’s theorem for a bounded operator is obtained; and in the general case of a Banach space the idea of equivalent semi-inner products is introduced. Applications are given to a class of initial boundary value problems in a bounded domain in $R^n $ under nonlinear perturbations.

Introduction. In this paper a series of propositions are given concerning commutative algebras of operators on a Banach space and more especially commutative algebras of scalar operators. A number of the results are known and due to W. G. Bade [1; 2] but different proofs are given here. The inspiration for this paper, both in the choice of the subject matter and of method, has largely been derived from [2; 7] and [10]. The material presented here is divided into four paragraphs. The first, which is introductory, contains various results on spectral families of measures. The principal propositions are contained in paragraphs 2 and 3. Theorem 1, proved in paragraph 2, is a generalisation of a theorem due to W. G. Bade [2], and almost all the other results of this same paragraph are more or less consequences of it. In paragraph 3 it is shown that, under certain conditions, an algebra of scalar operators can be identified, in a sense made precise below, with a von Neumann algebra. This fact makes it possible to reduce many results concerning algebras of scalar operators or a-complete boolean algebras of projections, in Banach spaces, to corresponding results in Hilbert spaces. Various remarks on spectral families of measures are made in paragraph 4.

In this article we present the basic facts of a general theory of linear operators in Hilbert spaces equipped with a Hermitian-bilinear metric. In particular, we investigate the spectra of operators of certain classes, most frequently encountered in applications.

We consider a two-point boundary value problem for the equation $u_{tt} - A(t)u = f(t,u)$ where, for each t, $A(t)$ is an unbounded operator in Hilbert space whose domain varies with t. We establish the existence (but not the uniqueness) of “weak” solutions of this problem by using finite difference methods. When $A(t)$ represents a differential operator in the space variables, the method of proof can be converted into a convergent numerical procedure.

CHANGING from a differential to a difference expression is naturally treated as a change from an operator, which acts in a functional space, to an operator, which is defined in a factor-space of this functional space by a subspace of functions which vanish at the nodes of the network. In the present paper a modification of the Banach-Steinhaus theorem on the boundedness of the sequence of operators is proved in the case where the operators are defined on a family of factor-spaces. This theorem is then used to investigate the convergence and stability of the difference method.

Let T T be a bounded linear operator on an L 1 {L_1} -space and τ \tau its linear modulus. It is proved that if the adjoint of τ \tau has a strictly positive subinvariant function then the following two conditions are equivalent: (i) T n {T^n} converges weakly; (ii) ( 1 / n ) Σ i = 1 n T k i (1/n)\Sigma _{i = 1}^n{T^{{k_i}}} converges strongly for any strictly increasing sequence k 1 , k 2 , ⋯ {k_1},{k_2}, \cdots of nonnegative integers.

(I) T = f 2. ES(d~)., moreover; (2) the residual spectrum of T is void, and E r is a weakly continuous ~) function of rectangles, (3) the set n(E T) of all complex t such that Er({1})4:O coincides with the point-spectrum of T. Suppose henceforth that ~ is not a Hilbert space. What can then be said about the properties (1)--(3) when T6 [~] ? The theories "of DUNFORD [2] and LORCH [12] are too restrictive to be applicable to the algebra s of "Laurent operators" [5]; r < I/p] (we suppose t < p < oc and p ~ 2). For example, consider the following two members of ~: the transformation H which is represented by the Hilbert matrix

Given a closed linear operator A A with dense domain in a Banach space X X , M. Schechter [4] utilized the Lebesgue integral to construct a family of bounded linear operators from X X to the Calderón complex interpolation space ( X , D ( A ) ) 8 {(X,D(A))_8} [2], where D ( A ) D(A) , the domain of A A in X X , is a Banach space under the norm \[ | | x | | D ( A ) = | | x | | + | | A x | | . ||x|{|_{D(A)}} = ||x|| + ||Ax||. \] In this paper we utilize the complex functional calculus, which provides a more natural setting, to construct a similar family of operators. At the same time we achieve a strengthening of the Schechter result, for in the proof of our theorem we make no use of the adjoint A ∗ {A^ \ast } of A A and consequently do not require the domain of A A to be dense in X X . A completely analogous procedure would permit the removal from the Schechter theorem, referred to above, of the hypothesis that the domain of A A is dense in X X .

The Fredholm Alternative Theorem gives the notion of bounded and continuous operator which makes a family F of elements of C[a, b] bounded and equicontinuous. A continuous operator of finite rank is completely continuous but every continuous operator is not completely continuous operator. Keywords: Arzela-Ascoli's theorem; completely continuous; equicontinuous and Fredholm alternative theorem DOI: http://dx.doi.org/10.3126/bibechana.v8i0.5700 BIBECHANA 2012; 8: 127-130

The article is a survey of work on non-linear monotone operators on Banach spaces. Let be an operator acting from a Banach space into its adjoint space. If on the whole space the scalar product inequality holds, then is said to be a monotone operator. It turns out that monotonicity, in conjunction with some other conditions, makes it possible to obtain existence theorems for solutions of operator equations. The results obtained have applications to boundary-value problems of partial differential equations, to differential equations in Banach spaces, and to integral equations.Here is a list of questions touched upon in the article. General properties of monotone operators. Existence theorems for solutions of equations with operators defined on the whole space or on an everywhere dense subset of the space. Fixed point principles. Approximate methods of solution of equations with monotone operators. Examples that illustrate the possibility of applying the methods of monotonicity to some problems of analysis. In conclusion, the article gives a bibliography of over one hundred papers.

The theory of abstract unbounded operators and their extensions developed along two main lines. The first is the theory of extensions of symmetric operators (Achiezer and Glasman [3]). In this case the problem is to extend a symmetric operator to a self-adjoint one. This theory includes necessary and sufficient conditions for the existence of such extensions, as well as complete characterization of the nature of these extensions, especially the nature of the spectrum of various extensions pertaining to the same operator. This theory imposes severe conditions and can be applied only in particular cases. In order to deal with abstract problems arising in the theory of partial differential equations and integral equations a relaxation is needed. The notions of "generalized solution," "weak solution," etc. (for example in Lax [1]) and the theory of distributions lead to the following problem of extension of general abstract operators. (For the sake of simplicity the illustration below will be confined to operators acting in Hilbert space.) The operator (differential, integral, etc.) !T is embedded in a certain space E. In this space a "natural" domain Do is considered. Denote Y acting on Do by To. To is in general bounded below on Do so it may be closed to an operator To ' To acting on Do D bo. By this process the "generalized solutions" are obtained. In most cases the range of To is not the whole space so an extension is needed. Among all possible extensions only the "weak solutions" are considered. They are obtained by looking at a natural "formal adjoint" 9* defined on the closure of the set bo. Denote this operator by T* and denote (T1*)* by T1. Any function in the domain of T1 is called a "weak solution." In general the range of T1 is the whole space but there are "too many" weak solutions because there are null solutions. So the problem is to find an intermediate To c T c T1 having a bounded inverse defined on the whole space. Somewhat different versions are in G. C. Rota [4] and Gohberg and Krein [5]. These general theories have only weak results, because of the generality of the operators considered.

Within the theory of regular perturbation the restriction on a source function is slightly relaxed so that the theory may include cases in which the interaction part of Hamiltonian not only has no dense domain common with the free part, but also it is never defined in a Hilbert space. A smallness condition of the interaction part expressed in terms of symmetric quadratic forms ensures the total Hamiltonian to be uniquely determined as a self-adjoint operator. All the results concerning the strong asymptotic limits of operators which have been established previously for more restrictive source functions are retained in these cases.

In this paper is presented a theory of unbounded operators which admit spectral distributions. The compactness of the support in the bounded case has been replaced by certain growth conditions at infinity. Properties of these operators are studied, including the existence of a Spectral Mapping Theorem. Inttoduction. In [2], Foias introduced the class of bounded operators in Banach spaces which in lieu of spectral measures admit spectral distributions. He proved certain basic properties of these "generalized scalars", such as the Spectral Mapping Thorem. Further investigations of this class of operators and of spectral distributions were made by various authors in [1], [2], [4], [5], [6],1 [8], and [10]. In [5], the present author studied the class of generalized scalars with real spectrum, which were called generalized pseudo-hermitian (g.p.h) operators. These operators were characterized by the elementary condition that the real exponential group generated by such an operator have polynomial growth in the uniform operator norm.2 In attempting to study the spectral properties of the infinitesimal generators of CO-groups with polynomial growth, the author has been led to consider unbounded generalized scalar operators. In this paper is presented a theory of unbounded spectral distributions and of the unbounded generalized scalar operators which they represent. The compactness of the support in the bounded case has been replaced by certain growth conditions at infinity sufficient to insure that the definitions are correct and the resulting class of operators well behaved, in that they generalize both the bounded generalized scalars and the unbounded normal operators in Hilbert space. Some properties of the class are discussed, culminating in a Spectral Mapping Theorem. 1. Basic definitions. DEFINITION 1. Let T be a mapping from the space -9(R2) of complex test functions into the space Y(X) of linear continuous operators on a Received by the editors April 6, 1971. 4MS 1970 subject classifications. Primary 47A65, 47A60, 47B40; Secondary 46F99. I Also considers unbounded operators. 2 Cf. Theorem 2 of [101. (? American Mathematical Society 1972