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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.

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.

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)

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.

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.

A theorem is proved on the inversion of operators of the form (d/dt) + A, where A is a linear operator in a space of vector functions bounded on the axes.

It will be shown that it is only necessary to test the field. operators in timelike direction. ∫A(x0,x)f (x0)dx0 exists as unbounded operator and is infinitely often difierentiable in spacelike direction.

The following Theorem 1 and Theorem 2 are established. The proof utilizes the elementary properties of symmetric operators. It is shown that the symmetry condition is necessary for these theorems to hold.

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.

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

Theorems of M. A. Krasnosel'skii are generalized which concern fixed points of completely continuous operators which compress or expand a cone, and a topological principle for fixed points in the case of P-compact operators.

Let A be the closed unbounded operator inLp(G) that is associated with an elliptic boundary value problem for a bounded domainG. We prove the existence of a spectral projectionE determined by the set Γ = {λ;θ1≦argλ≦θ2} and show thatAE is the infinitesimal generator of an analytic semigroup provided that the following conditions hold: 1

We give an extension of the Riesz-Thorin theorem to linear operators which map Morrey spaces in spaces of the same or similar type.

(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

The definition given by P. P. Korovkin of operators of the class Sm and conditions for the convergence of these operators to the identity operator are extended to apply to regular operators from a K-space R0 with a unit, into a K-space R1 where R0 and R1 are normally contained in the union of the spaces S [a, b] and s.

Generailzed Green's functions are kernels of integral operators with certain properties. Solution operators with these properties are calledGreen's operators. Necessary and sufficient conditions for the existence of Green's operators are given for the general case, and for operators on Banach spaces.

The famous work of O. Perron [19] and G. Frobenius [5-7] on matrices with positive and nonnegative elements is still of high interest in modem mathematics. The mentioned papers have given rise to many generalizations of linear transformations in abstract linear spaces, e.g., the results of Birkhoff [1], Bonsall [2-4], Karlin [10], Krasnoselskii [11], Krein-Rutman [12], Niiro-Sawashima [18], Sawashima [20], Schaefer [23, 25, 27] and other authors. Most of the papers devoted to the Perron-Frobenius theory of positive linear transformations are concerned mainly with the existence problems. Recently H. Schaefer [21, 24, 26, 28] and P. Lotz [13] considered another part of the Frobenius theory which was the description of the peripheral spectrum of a positive linear irreducible transformation in a Banach lattice. In the present paper an attempt is made to complete the generalized Frobenius theory by those statements in which the spectral radii of two or more positive linear transformations in a Banach space are compared. The results are based on a general minimax principle which might be of autonomous interest. As an application of the comparison theory, some results are given concerning the rate of convergence of certain classes of iteration processes. Another type of application is the problem of criticality of one-parameter systems of positive linear transformations (see [15]). It should be remarked that for our comparison theory the existence of certain proper elements is needed. The existence of such proper elements is guaranteed, as a rule, by certain compactness properties of either the intersection of the unit ball B and the positive cone K, or its image T(B n K) by the considered map T. We shall not repeat the compactness assumptions guaranteeing the existence of the required proper elements. We shall assume only that appropriate eigenelements exist. This means that our comparison theory is based on the existence of the mentioned proper elements and formally not on the compactness of B or TB. Up to now, however, most of the existence proofs have depended on these compactness assumptions. Similarly, like the existence results, our comparison theory carries over to bounded linear positive endomorphisms of partially ordered quasi-complete locally convex spaces.