Showing papers on "Nuclear operator published in 1975"

Integral operators in spaces of summable functions

Abstract: Contents Preface xm Chapter 1. Linear operators in L a spaces 1 1. The space L x 1 1.1. Description of the spaces 1.2. Criteria for compactness 1.3. Continuous linear functionals and weak convergence 1.4. Semi-ordering in the spaces S and L x 1.5. Projections and bases of Haar type 1.6. Operators in the spaces L a

Shorted operators. ii

Abstract: For a positive operator A acting on a Hilbert space, the shorted operator $\mathcal{L}( A )$ is defined to be the supremum of all positive operators which are smaller than A and which have range lying in a fixed subspace S. This maximization problem arises naturally in electrical network theory. In this paper we prove that the shorted operator exists, and develop various properties, including a relation to parallel addition [Anderson and Duffin, J. Math. Anal. Appl., 11 (1969), pp. 576–594]. The basic properties of the shorted operator were developed for finite-dimensional spaces by Anderson [this Journal, 20 (1971), pp. 520–525] ; some of these theorems remain true in all Hilbert spaces, but the proofs are different.

Spaces of compact operators and their dual spaces

Abstract: LetE andF be reflexive Banach spaces andC the space of all compact linear operators fromE toF. A representation of the dual space ofC is given and it is proved thatC is either reflexive or nonconjugate. Applications of these results are also given.

Some linear topological properties of the spaces $C_p$ of operators on Hilbert space

Abstract: © Foundation Compositio Mathematica, 1975, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

An inequality among determinants

Abstract: We prove that for any trace class operators, A,B, det (1+|A+B|) ≤ det (1+|A|) det (1+|B|) where |C| = (C*C)1/2.

Some operator algebras in nested Hilbert spaces

Abstract: The existence of a von Neumann algebra of operators and hence the existence of projections acting in any nested Hilbert space is proved. Some other algebras of operators are studied. All those algebras are exhibited in a particular class of Nested Hilbert Spaces, namely sequence spaces.

Compact operators in the algebra generated by essentially unitary ₀ operators

Abstract: It will be shown that the compact operators in the weakly closed algebra generated by an essentially unitary CO contraction are weakly dense in the algebra. The result implies the extension of a double dual theorem of Kriete, Moore and Page and yields a partial answer to a question on reductive algebras raised by Rosenthal. A contraction T on a separable Hilbert space is called essentially unitary in case both 1 T*T and 1 TT* are compact. A technique, based on Muhly's characterization [6] of compact operators commuting with a COO operator, was introduced in [5] for showing that the weakly closed algebra 2T generated by an essentially unitary CO contraction T and the identity contains nonzero compact operators. That technique will be used here to show that the identity, and hence every operator in 21T is in fact the weak limit of a sequence of compact operators in ET. Two consequences of this result are to be derived. Kriete, Moore and Page [4] showed that if T is the compression of a simple shift operator to the orthogonal complement of one of its nontrivial invariant subspaces, then the commutant S1' of T may be identified with the second dual of the Banach space of compact operators in 2L1. (Actually their theorem is more general as it deals with intertwining operators between pairs of such compressions.) Since a compression of the simple shift to the orthogonal complement of one of its nontrivial invariant subspaces is a CO operator (see [9]) whose defect operators are rank one, we see that the result of Kriete, Moore and Page is a special case of Corollary 1 below in which T is merely required to be an essentially unitary CO contraction. Thus, as they conjectured, considerable generalization of their result is possible. Received by the editors May 23, 1974. AMS (MOS) subject classifications (1970). Primary 47C05.

Selfadjoint differential operators with an infinite number of independent variables

Abstract: In this paper a new method of definition of differential operators with an infinite number of independent variables is proposed, and an analysis of the selfadjointness in the corresponding Hilbert spaces is carried out. Certain spectral properties of such operators are also investigated. In contrast with previous analysis, the spaces of functions of an infinite number of variables in which the operators being studied act are, in general, not infinite tensor products of spaces of functions of a finite number of variables.Bibliography: 5 items.

Properties of interpolation spaces of completely continuous operators

Abstract: We consider interpolation spacesG θ,p between the spaceG 1 of nuclear operators and the spaceG ∞ of completely continuous operators in a Hilbert space. We obtain an exact expression for the norm of an operator ofG θ,p by means of its approximation numbers. We consider the question of separability of the symmetric normed ideals ofG θ,p , and the question of the duals to these s.n. ideals.

Operators satisfying certain growth conditions. ii

Abstract: It is proved that the condition wp((T - zl)~l) = \/d(z, o(T)), w (•) being the operator radius of Holbrook, implies the existence of certain eigenvalues and normal eigenvalues for a Hubert space operator T. This extends known results based on a norm condition (p = 1) and allows a similar extension of various consequences of these results.