A Study on Semiclosed Subspaces and Semiclosed Operators in Hilbert Spaces

Abstract

Semiclosed subspaces (para-closed subspaces, in the terminology of C. Fioas) of Hilbert spaces have been considered for a long time, as a more flexible substitute of closed subspaces of Hilbert spaces. What is even more interesting is that the notion of semiclosed subspace coincides with that of a Hilbert space continuously embedded in H. It is proved that the collection of all Hilbert spaces continuously embedded in a given Hilbert space H is in a bijective correspondence with the convex cone of all bounded positive self-adjoint operators in H. For two bounded operators A and B in H with the kernel condition N(A) ⊆ N(B), the quotient [B=A] defined in Izumino (1989), by Ax ! Bx, x 2 H. A quotient of bounded operators so defined is what was introduced by Kaufman (1978), as a \semiclosed operator", and several characterizations of it are given. It is proved that the family of quotients contains all closed operators and is itself closed under \sum" and \product". A merit for the quotient representation of a semiclosed operator is to make use of the theory of bounded operators. In the thesis, semiclosed subspaces and semiclosed operators in Hilbert spaces have been studied extensively.