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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.

A self-adjoint second-order differential expression with an unbounded operator coefficient is considered in the space of vector functions. The domain of definition of the minimum and maximum operators generated by this expression is investigated, it is shown that any generalized resolvent of the minimum operator is an integral operator, and expansion in eigenfunctions is performed.

Unbounded selfadjoint representations of ∗ ^\ast -algebras are studied. It is shown that a selfadjoint representation of the enveloping algebra of a Lie algebra can be exponentiated to give a strongly continuous unitary representation of the simply connected Lie group if and only if the representation preserves a certain order structure. This result follows from a generalization of a theorem of Arveson concerning the extensions of completely positive maps of C ∗ {C^ \ast } -algebras. Also with the aid of this generalization of Arveson’s theorem it is shown that an operator π ( A ) ¯ \overline {\pi (A)} is affiliated with the commutant π ( A ) ′ \pi (\mathcal {A})’ of a selfadjoint representation π \pi of a ∗ ^\ast -algebra A \mathcal {A} , with A = A ∗ ∈ A A = {A^ \ast } \in \mathcal {A} , if and only if π \pi preserves a certain order structure associated with A and A \mathcal {A} . This result is then applied to obtain a characterization of standard representations of commutative ∗ ^\ast -algebras in terms of an order structure.

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 A and B be closed operators on Banach spaces X and Y. Assume that A and B have nonempty resolvent sets and that the spectra of A and B are unbounded. Let α be a uniform cross norm on X ⊗ Y. Using the Gelfand theory and resolvent algebra techniques, a spectral mapping theorem is proven for a certain class of rational functions of A and B. The class of admissable rational functions (including polynomials) depends on the spectra of A and B. The theory is applied to the cases A ⊗ I + I ⊗ B and A ⊗ B where A and B are the generators of bounded holomorphic semigroups.

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.

We introduce a class of linear operators, containing the positive operators, in which P. P. Korovkin's theorem on the conditions and order of convergence of positive operators hold. We consider also the class of linear operators, convergent to the derivative, in which similar theorems hold.

Fort ∈ [a, b], letA(t) be the unbounded operator inH 0,p (G) associated with an elliptic-boundary value problem that satisfies Agmon’s conditions on the rays λ=±iτ, τ ≥0. Existence and uniqueness results are obtained for weak and strict solutions of two-point problems of the type (du/dt)−A(t) u(t) =f(t),E 1(α)u (α)=u α,E 2 (β)u (β)=u β. Here [α, β) χ- [a, b],E 1 (α) andE 2 (β) are spectral projections associated withA(α) andA(β) respectively, andA(α)E 1 (α) and =A (β)E 2 (β) are infinitesimal generators of analytic semigroups. WhenA(t) andf(t) are analytic in a convex, complex neighborhoodO of [a, b] we show that for someθ i ,i=1,2, any solution ofdu/dt =A(t)u (t)=f(t) in [a, b] is analytic and satisfies the above equation in the setO∩{t; t ≠ a, t ≠ b, | arg (t −a)

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\

Consider the equation (i) (da/dt)—A(t)u(t)=f(t) where fort ∈ [a, b],A(t) is a densely defined and closed linear operator in a Banach spaceX. Assume the existence of bounded projectionsE i(t),i=1, 2, such thatA(t) E 1(t) and —A(t)E 2(t) are infinitesimal generators of analytic semigroups andA(t) is completely reduced by the direct sum decompositionX = Σ b = 1/2 ⊕E i (t)X. We show that any solutionu(t) of (i) is inC ∞(a, b) and satisfies the inequalities (1.2) provided thatf(t) andA(t) are infinitely differentiable in [a, b] in a suitable sense. In caseA(t) andf(t) are in a Gevrey class determined by the constants {M n} we have (1. 3). Applications are given to the study of solution of (i) where fort ∈ [a, b]A(t) is the unbounded operator inH 0,p (G) associated with an elliptic boundary value problem that satisfies Agmon’s conditions on the rays λ=±iτ, τ > 0.

1. INTR~DLJcT~~N The connection between the convergence of the sequence {(I - aA*A)nj, where A is a bounded linear operator on a Hilbert space H, and the iterative computation of the pseudoinverse A+ of A has been investigated by various authors [2, 19, 20, 22 and 251. Also, the convergence of the sequence {T”}, where T is a bounded linear operator on a Banach space X, has received much attention over a period of years mostly in connection with the iterativp solution of linear equations [2, 8,9, 14, 15 and 181. The sequence {T”} can be replaced by (u,(T)), h w ere each a,, is an nth degree polynomial whose coefficients are nonnegative and sum to 1 [5, 6, 16 and 171. The convergence of {T”) has also been used as a hypothesis for theorems on series represents tion of Banachspace pseudoinverses [31, 19 and 201. The purpose of this paper is twofold: To study general properties of power convergent operators (bounded linear operators on X for which {Tn) con verges in a suitable operator topology), and to apply these results to the series representation of Banachspace pseudoinverses. In Section 2, we survey some properties of power convergent operators [15, 161, and present some new results and examples on spectral analysis of these operators. Section 3 gives some necessary and sufficient conditions for power convergence of certain classes of mostly Banach-space operators, Section 4 then presents conditions of the Stein type for power convergence of Hilbert-space operators. In Section 5 we give a definition of the pseudoinverse for bounded linear operators on Banach space with arbitrary range. This definition seems to be the most natural generalization of the classical Hilbert-space pseudoinverse for operators with arbitrary ranges [12, 19,20 and 251, and agrees with the definition given in [l] and [31] in the case that the ranges are closed. We then give series representations for the pseudoinverse and partial inverse; these results have numerous applications to the iterative solution of linear operator equa446

On a Banach space the numerical range of an unbounded operator has a certain density property in the scalar field. Consequently all hermitian and dissipative operators are bounded. For a smooth or separable or reflexive Banach space the numerical range of an unbounded operator is dense in the scalar field.

We present some results concerning the general theory of Banach ideals of operators and give several applications to Banach space theory. We give, in Section 3, new proofs of several recent results, as well as new operator characterizations of the L p -spaces of Lindenstrauss and Pelczynski. In Section 4 we prove that the space of absolutely summing operators from E to F is reflexive if both E and F are reflexive and E has the approximation property. Section 5 concerns Hilbert spaces. In particular, we compute the relative projection constant of Hilbert spaces in L p (μ)-spaces.

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.

On the half-line we investigate the following equation in a Banach space: (1)where are closed operators which commute with . We consider the following classes of equations: parabolic, inverse parabolic, hyperbolic, quasi-elliptic, and quasi-hyperbolic. We present boundary value problems for these classes and prove that they are well-posed. The proofs are based on a solvability theorem for the operator equation , where is a closed operator.Bibliography: 20 items.

The perturbation theory of Hille and Phillips for semigroups of bounded linear operators on a Banach space is modified to apply to the semigroups of positive traceclass operators encountered in quantum statistical mechanics.

Representations of bounded linear operators on Banach function spaces of vector-valued functions to Banach spaces are given in terms of operator-valued measures. Then spaces whose duals are Banach function spaces are characterized. With this last information, reflexivity of this type of space is discussed. Finally, the structure of compact operators on these spaces is studied, and an observation is made on the approximation problem in this context.

Let E E be a Banach space and let A A be a continuous function from E E into E E . Sufficient conditions are given to insure that the differential equation u ′ ( t ) = A u ( t ) u’(t) = Au(t) has a unique solution on [ 0 , ∞ ) [0,\infty ) for each initial value in E E . One consequence of this result is that if − A -A is monotonic, then − A -A is m m -monotonic and A A is the generator of a nonexpansive semigroup of operators.

Extension of pseudo-inverse operator theory to Hilbert space operators that are unbounded and have arbitrary range

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 ].