Abstract: A survey of distribution theory.- Distributions with rational singularities.- Finitely generated pseudo-differential operators.- On the laplace comparison algebras for L2(?n).- Elliptic boundary problems.

Abstract: This volume describes the theory and the various uses of functional equations mainly in view of functional analysis.

Abstract: This survey is devoted to an exposition of results on the asymptotics of the discrete spectrum of self-adjoint differential operators, mainly partial differential operators.

Abstract: Two algebras of global pseudo-differential operators over ℝn are investigated, with corresponding classes of symbols A0=CB∞ (all (x, ξ)-derivatives bounded over ℝ2n), and A1 (all finite applications of ∂xj, ∂ξj, and epq=ξp∂ξq−p∂xp on the symbol are in A0). The class A1 consists of classical symbols, i.e., ∂ α x ∂ β ξ a= 0((1+|ξ|)−|α|) for x ∈ Kc ℝ;n, K, compact. It is shown that a bounded operator A of 210C=L2(Rn) is a pseudo-differential operator with symbol a∈Aj if and only if the map A→G−1AG, G∈ gj is infinitely differentiable, from a certain Lie-group gj c GL(210C) to ℒ(210C) with operator norm. g0 is the Weyl (or Heisenberg) group. Extensions to operators of arbitrary order are discussed. Applications to follow in a subsequent paper.

Abstract: The main theme of this paper is a study in some detail of Banach ideals of continuous linear operators between Banach spaces factoring compactly through lp (1≤p<∞) or co, called p-compact and ∞-compact operators respectively. Recently operators of these types have been studied in [4] within the framework of locally convex spaces which are dense subspaces of p-compact projective limits of Banach spaces. These ideals show close resemblance to the ideals of p-nuclear operators-for the case p=∞ they coincide. Analogously to results of Grothendieck concerning continuous linear operators, we consider vector sequence spaces isometric isomorphic to certain spaces of compact linear operators. A representation theorem for p-compact operators is deduced and isometric properties of the ideal norm are treated. The paper also includes some remarks on unconditional convergence and related operator ideals and a representation for the complete ɛ-tensor product $$\ell^{P}\tilde{\bigotimes}_{\varepsilon}E$$ (1≤p<∞) is given.

Abstract: In this paper, we study the factorization and the representation of Fredholm operators belonging to the algebra $\mathcal{R}$ generated by inversion and composition of Toeplitz integral operators. The operators in $\mathcal{R}$ have the interesting property of being close to Toeplitz (in a sense quantifiable by an integer-valued index $\alpha $) and, at the same time, of being dense in the space of arbitrary kernels. By using these properties, we derive a set of efficient algorithms (generalized fast-Cholesky and Levinson recursions) for the factorization and the inversion of arbitrary Fredholm operators. The computational burden of these algorithms depends on how close (as measured by the index $\alpha $) these operators are to being Toeplitz.We also obtain several important representation theorems for the decomposition of operators in $\mathcal{R}$ in terms of sums of products of lower times upper triangular Toeplitz operators. These results can be used to approximate operators corresponding to noncausa...

Abstract: In this paper, we study the factorization and the representation of Fredholm operators belonging to the algebra R generated by inversion and composition of Toeplitz integral operators. The operators in R have the interesting property of being close to Toeplitz (in a sense quantifiable by an integer-valued index a) and, at the same time, of being dense in the space of arbitrary kernels. By using these properties, we derive a set of efficient algorithms (generalized fast-Cholesky and Levinson recursions) for the factorization and the inversion of arbitrary Fredholm operators. The computational burden of these algorithms depends on how close (as measured by the index a) these operators are to being Toeplitz. We also obtain several important representation theorems for the decomposition of operators in R in terms of sums of products of lower times upper triangular Toeplitz operators. These results can be used to approximate operators corresponding to noncausal and time-variant systems in terms of operators represent- ing causal and anticausal time-invariant systems, a property that has a large number of potential applications in signal processing problems.

Abstract: In a Hilbert space H, an operator C is semiclosed provided that there exists a bounded operator B on H, with range the domain of C, such that CB is bounded. The family of all such operators in H is the smallest family containing all closed operators and itself closed under any one of the following: (1) sums, (2) products, (3) strong limits on domains of closed operators. In fact, every algebraic combination of closed operators in H is the sum of two closed one-to-one operators with the same domain and closed ranges. Introduction. Suppose that (H, ) is a complete complex inner product space (i.e., a Hilbert space) and let (H x H, -) denote the usual product space. The statement that S is a semiclosed subspace of H means that S is a linear (but not necessarily closed) subspace of H for which there is an inner product ' such that (S, ') is complete and continuously included in H, in the sense that there is a nonnegative number b such that ' for all x in S. Now, an operator in H is understood to be a linear function from a linear subspace of H into H; hence we propose the following: DEFINITION. A semiclosed operator in H is an operator in H which, as a subspace of H x H, is semiclosed. Evidently each closed operator in H is semiclosed, since any closed subspace is semiclosed. However, there exists a semiclosed operator in H whose closure is all of H x H (see ?2). In fact, every function from a countable linearly independent subset of H into H has an extension which is a semiclosed linear operator in H (Theorem 12). Herein, we show that the family S C(H) of all semiclosed operators in H is closed under addition, multiplication, inversion, restriction to semiclosed subspaces of H, and strong limits in H on such subspaces. Moreover, we find each of the following to be a necessary and sufficient condition on an operator C in H in order that C be in S C(H) (let S denote the domain of C): (I) C is the sum of two closed reversible (one-to-one) operators in H each having domain S and closed range (hence, continuous inverses). (II) C is the product of a continuous operator on H with a closed positive Presented to the Society, January 24, 1979; received by the editors September 8, 1978. AMS (MOS) subject classifications (1970). Primary 47A45, 47A65.

Abstract: We prove that are positive-definite Hermitian matrices and where V denotes the supremura in the sense of spectral order. Actually the convergence is proved (in the sense of strung convergence) in the case when the As are bounded operators on Hilbert space. Generalization to unbounded operators is briefly discussed

Abstract: In this paper, we describe the symbol calculus and index theorem for Wiener-Hopf operators on the group of complex 2×2 unitary matrices U2.

Abstract: In this note, we examine some of the properties of Hermitian operators on complex unital Banach Jordan algebras, that is, those operators with real numerical range. Recall that a unital Banach Jordan algebra J, is a (real or complex) Jordan algebra with product a ˚ b, having a unit 1, and a norm ∥·∥, such that J, with norm ∥·∥, is a Banach space, ∥1∥ = 1, and, for all a and b in j,

Abstract: Multilinear techniques are used to characterize unitary matrices in terms of a generalized numerical range. This characterization is then applied to analyze the structure of all linear operators on matrices which preserve this numerical range. The results generalize V. J. Pellegrini's determination of all linear operators preserving the classical numerical range.

Abstract: This paper contains two new characterizations of generators of analytic semigroups of linear operators in a Banach space. These characterizations do not require use of complex numbers. One is used to give a new proof that strongly elliptic second order partial differential operators generate analytic semigroups inL p , 1