Johnson, P. SamBalaji, S2020-08-052020-08-052014https://idr.nitk.ac.in/handle/123456789/14387Semiclosed 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.enDepartment of Mathematical and Computational SciencesSemiclosed subspaceoperator rangeinvariant subspacesemiclosed operatorquotient of bounded operatorsclosed rangeHyers-Ulam stabilityThesis