Showing papers on "Nuclear operator published in 1976"

Analysis with weak trace ideals and the number of bound states of Schrödinger operators

Abstract: We discuss interpolation theory for the operator ideals Ip p defined on a separable Hilbert space as those operators A whose singular values

Semigroup of operators on dual Banach spaces

Abstract: In this paper, we give a short and simple proof to a more general version of a recent result of Yeadon for semigroups of weak*-continuous operators on a dual Banach space. Our result has application to amenable groups and property P of a von Neumann algebra.

Commuting well-bounded operators on Hilbert spaces

Abstract: A bounded linear operator T on a complex reflexive Banach space is said to be well-bounded if it is possible to choose a compact interval J = [a, b] and a positive constant M such thatfor every complex polynomial p, where ‖p‖J denotes sup {|p(t)|:t ∈ J}. Such operators were introduced and first studied by Smart (4). They are of interest principally because they admit (and in fact are characterised by) an integral representation similar to, but in general weaker than, the integral representation of a self-adjoint operator on a Hilbert space. (See (2) and (4) for details.) It is easily seen, by verifying (1) directly, that T is well-bounded if it is a scalar-type spectral operator with real spectrum.

One dimensional perturbations of compact operators

Abstract: Let A be a compact selfadjoint operator acting on a Hilbert space H. P denotes a one dimensional projection also acting on H. It is shown that the eigenvalues of A and A + tP (t > 0) interlace on the real axis. A converse of this result is also proved. The purpose of this note is to study the change which results in the spectrum of a compact selfadjoint operator acting on a Hilbert space H by the addition to it of a positive multiple of a one dimensional projection. We begin by stating a theorem of Hochstadt [1] and giving a variant of his proof based on [3] depending on a study of the resolvents of the operators involved, and our methods lead us naturally to a converse to this result. Let A he a compact selfadjoint operator acting on a Hilbert space H. P denotes a one dimensional projection in the direction of a normalised element x of H: then Py = iy,x)x for every y in H. B is the operator A + tP, t > 0. Theorem 1. Suppose that the null space of A is empty. Between every pair of distinct, successive eigenvalues IXi,Xi+X) there is precisely one eigenvalue of B in one of the intervals (a,,X, + 1], [a,,a,+1) or (A,-,A, + 1). Proof. We can associate with A [2] a complete orthonormal set {d>,} such that each (#>, is an eigenvalue of A so that Ai = A; + t(v,x)x, and applying the resolvent Rc, we obtain RcU = v + tiv,x)Rcx and v = Rcu = Rtu tiv,x)Rcx. Taking the inner product with x, iv,x) = IRcu,x) tiv,x)IRcx,x). Solving for (i/, x), we obtain Received by the editors January 13, 1975 and, in revised form, March 4, 1975 and April 18, 1975. AMS (MOS) subject classifications (1970). Primary 47A55, 47B05; Secondary 34B25. © American Mathematical Society 1976 58 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ONE DIMENSIONAL PERTURBATIONS OF COMPACT OPERATORS 59 (v,x) (Rsu,x)/(l + t(R{x,x)). Substituting for (v, x), we obtain t(Rcu,x) R'tU = RrU --f--rRyX. * f 1 + t(Rix,x) s

On the Range of the sum of Nonlinear Operators

Abstract: Publisher Summary This chapter discusses the range of the sum of nonlinear operators, mappings in infinite dimensional spaces and some of its applications, monotone operators, linear (unbounded) operator, amonotone hemicontinuous operator, and various theorems.

Aspects of bounded perturbation theory

Abstract: 1. Abstract This paper is concerned with the stability of certain properties of linear operators in locally convex topological vector spaces under perturbations by operators which are small in some sense. Section 3 deals with the very useful concept of Banach balls which was introduced by Raĭkov [9]. Some properties are discussed. The following section investigates the invertibility of certain operators generalizing results of Robert [10] and de Bruyn [2],[3]. These results are used extensively in the sequel. We go on to discuss Riesz operators. We obtain results stronger than those of de Bruyn [1] with regard to asymptotically quasi-compact operators in locally convex spaces. The proofs are basically adaptations of those from [1]. In the final section we observe some results concerning the range ad null space of an operator perturbed by bounded operators. We obtain a result very similar to an unproved theorem of Vladimirskiĭ [a] and point out their differences. MOS codes 4601, 4710, 4745, 4768, 4755. Thi...

On a class of singular integral operators with kernels of polar-logarithmic type

Abstract: Integral operators with polar-logarithmic kernels which arise in connection with a problem of F D Gahov are studied in the space Necessary and sufficient conditions that the operator be Noetherian and a formula for the index of the operators are obtained A complete solution of Gahov's problem is obtainedBibliography: 10 titles

Estimates for the ∂-Neumann operator in weighted Hilbert spaces

Abstract: Estimates for the I operator are used to derive estimates for the Neumann operator in weighted Hilbert spaces The technique is similar to that used to prove regularity of solutions of elliptic partial differential equations A priori estimates are first obtained for smooth compactly supported forms and these estimates are then extended by suitable approximation results These estimates are applied to give new bounds for the reproducing kernels in the subspaces of entire functions

Joins and intersections of ideals of compact operators

Abstract: It is shown that certain well-known ideals of compact operators are the intersection of a decreasing, countable family of strictly larger ideals. Also, it is shown that if Tj and T2 are compact operators, neither of which lies in the principal ideal generated by the other, and if I is an arbitrary countably generated ideal, then there exist ideals $, and %2 such that % S $1 v $2and T i %i,,i = 1, 2.

Indecomposable Hilbert-Schmidt operators

Abstract: In 1973, L. G. Brown, R. G. Douglas, and P. A. Fillmore characterized the set of all operators of the form N + K where N is a normal operator and K is a compact operator and they asked whether or not every Hilbert-Schmidt operator is the sum of a normal operator and a trace class operator. They later asked if, for every Hilbert-Schmidt operator A, there exists a normal operator N for which A (D N is the sum of a normal operator and a trace class operator. We produce a large class of HilbertSchmidt operators A none of which is the sum of a normal operator and a trace class operator, and furthermore, for each arbitrary operator Q, A (D Q is not the sum of a normal operator and a trace class operator. We then use this to show that their characterization of the operators N + K does not hold true if we replace the class of compact operators by the trace class or by any ideal I for which I # I 1/2. In the case of the trace class, we show that even if the vanishing of the Helton and Howe trace invariant were added to the hypothesis of their characterization, it would not hold true. Let H be a separable, infinite-dimensional complex Hilbert space. Let L(H) denote the algebra of all bounded linear operators on H, and let K(H) denote the two-sided ideal in L(H) of all compact operators. Furthermore, let C2 and Cl be the Hilbert-Schmidt and trace class ideals, respectively, of compact operators in L(H), and let (N) be the class of normal operators in L(H). Finally, for each bounded operator A, let ae(A) denote the essential spectrum of A. L. G. Brown, R. G. Douglas, and P. A. Fillmore characterized (N) + K(H) by proving [2, Theorem 11.2] that an operator A is decomposable into the sum of a normal operator and a compact operator (i.e. A E (N) + K(H)) if and only if A*A AA* E K(H) and index(A l) = 0 for every X Z aJ(A). One then asks under what circumstances the ideal of compact operators can be replaced by the ideal Cl in this result. Indeed, if A E (N) + C1, then A not only satisfies the two conditions: (1)A*A AA* E Cl and (2) index(A AI) = 0 for every X X ae(A), but in addition, the trace of A *A AA* is clearly 0. In fact, more must be true. If A E (N) + C1, then the Helton and Howe trace invariant [4] vanishes for A. This follows since if Presented to the Society, January 25, 1975; received by the editors October 17, 1974 and, in revised form, July 28, 1975. AMS (MOS) subject classifications (1970). Primary 47A55, 47B05, 47B10; Secondary 47B47, 15A60, 47A65.

Structure theorem for a-compact operators

Abstract: A contraction T defined on a complex Hilbert space is called A- compact if there exists a nonzero functionf analytic in the open unit disc and continuous on the closed disc such that A(7) is a compact operator. In this paper, the factorization of f is used to obtain a structure theorem which describes the spectrum of T. Introduction. A bounded linear operator T on a complex Banach space X is called polynomially compact if there is a nonzero polynomial p(z) such that p(T) is compact. The spectrum of polynomially compact operators T has been completely described by F. Gilfeather (4) by showing that T is a finite sum of power compact operators. A similar problem of describing the spectrum arises in case p(z) is replaced by the uniform limit of polynomials. More explicitly, let T be a contraction defined on a Hilbert space, and A be the uniform closure of polynomials in the supremum norm over D; D is the open unit disc. The elements of A are analytic functions in D which have continuous extension to

On the imaginary component of a dissipative operator with slowly increasing resolvent

Abstract: We consider the class (MR 40 #6286) of bounded dissipative operators with real spectrum acting in the infinite-dimensional separable Hilbert space whose resolvents satisfy the following growth condition: Principal results: 1. The operator is the imaginary component of an operator (i.e., ) if and only if 0 is either an eigenvalue of infinite multiplicity for or a limit point for the spectrum of . 2. All linear operators with imaginary component and real spectrum belong to the class if and only if is nuclear: . Bibliography: 10 titles.

Self-conjugacy of abstract differential operators of the hyperbolic type

Abstract: Sufficient conditions are obtained for the self-conjugacy of certain operators generated on a semiaxis or a complete axis by a differential expression of the form l[y]=y″+Ay−q(t)y, where A is a self-conjugate operator bounded below in a separable Hilbert space H, and, for almost all t, q(t) is a bounded self-conjugate operator in H, locally summable with the square of the norm.

Some Degenerated Differential Operators on Vector Bundles

Abstract: Publisher Summary This chapter focuses on some degenerated differential operators on vector bundles; and discusses the notation of the mathematical concepts that are used in the described differential operators, and relations between the class of these differential operators and infinitesimal generators of semigroups of operators. It discusses k-th order differential operator, some auxiliary spaces, Hilbert space, Riemannian structure, the sesqui-linear form, a family of continuous sesquil-linear forms, bijections associated with certain differential operator, analytic theory of semi-groups of operators, method for the construction of semi-groups, Hermitian structure, various theorems, and lemmas.