On the invertibility of Hilbert space idempotents

This note is to present some results on the group invertibility of linear combinations of idempotents when the difference of two idempotents is group invertible.     

On complementary subspaces of Hilbert space

Every pair of non-trivial topologically complementary subspaces of a Hilbert space is unitarily equivalent to a pair of the form on a Hilbert space . Here is possibly , is a positive injective contraction and denotes the graph of . For such a pair the following are equivalent: (i) is similar to a pair in generic position; (ii) and have a common algebraic complement; (iii) is similar to for some operators on a Hilbert space. These conditions need not be satisfied. A second example is given (the first due to T. Kato), involving only compact operators, of a double triangle subspace lattice which is not similar to any operator double triangle.

     

On the infimum problem of Hilbert space effects

The quantum effects for a physical system can be described by the set ε(H) of positive operators on a complex Hilbert space H that are bounded above by the identity operator I. The infimum problem of Hilbert space effects is to find under what condition the infimum A∧B exists for two quantum effects A and B∈ε(H). The problem has been studied in different contexts by R. Kadison, S. Gudder, M. Moreland, and T. Ando. In this note, using the method of the spectral theory of operators, we give a complete answer of the infimum problem. The characterizations of the existence of infimum A∧B for two effects A. B∈ε(H) are established.     

On the definition of a probabilistic normed space

Summary In this paper we give a new definition of a probabilistic normed space. This definition, which is based on a characterization of normed spaces by means of a betweenness relation, includes the earlier definition of A. N. erstnev as a special case and leads naturally to the definition of the principal class of probabilistic normed spaces, the Menger spaces.     

On typical properties of Hilbert space operators

We study the typical behavior of bounded linear operators on infinite-dimensional complex separable Hilbert spaces in the norm, strong-star, strong, weak polynomial and weak topologies. In particular, we investigate typical spectral properties, the problem of unitary equivalence of typical operators, and their embeddability into C ₀-semigroups. Our results provide information on the applicability of Baire category methods in the theory of Hilbert space operators.     

On the infinite product of operators in Hilbert space

We give a necessary and sufficient condition for a certain set of infinite products of linear operators to be zero. We shall investigate also the case when this set of infinite products converges to a non-zero operator. The main device in these results is a weighted version of the König Lemma for infinite trees in graph theory.

     

On the irreducibility of the Hilbert scheme of space curves

Denote by the Hilbert scheme parametrizing smooth irreducible complex curves of degree and genus embedded in . In 1921 Severi claimed that is irreducible if . As it has turned out in recent years, the conjecture is true for and , while for it is incorrect. We prove that , and are irreducible, provided that , and , correspondingly. This augments the results obtained previously by Ein (1986), (1987) and by Keem and Kim (1992).

     

On unboundedly divisible distributions in Hilbert space

We establish a connection between the existence of integrals with respect to unboundedly divisible probability distributions in Hilbert space and the existence of integrals of functions from a certain class of functions with respect to their spectral measures. The results obtained in the present note constitute a strengthening of the propositions advanced by the author in [1].Translated from Matematicheskie Zametki, Vol. 16, No. 4, pp. 585–594, October, 1974.     

On the eigenvalues of compact operators in a Hilbert space

Summary This paper contains several remarks concerning the behavior of the eigenvalues of compact operators in a Hilbert space. The results are related to the theorem for a finite matrix obtained as Theorem 1 in my previous paper[5].This work was partially supported by a research grant of the Sakkokai Foundation.