A general construction of transmutation operators is developed for selfadjoint operators in Gelfand triples. Theorems regarding analyticity of generalized eigenfunctions and Paley-Wiener properties are proved.

We use elementary algebraic properties of left, right multiplication operators to prove some deep structural left $m$-invertible, $m$-isometric, $m$-selfadjoint and other related classes Banach space operators, often adding value extant results.

We extend to infinite dimensional separable Hilbert spaces the Schur convexity property 13 of eigenvalues of a symmetric matrix with real entries. Our framework includes both the case of linear, selfadjoint, compact operators, and that of linear selfadjoint operators 15 that can be approximated by operators of finite rank and having a countable family of eigenvalues. The abstract results of the...

For a class of bounded random selfadjoint operators (Lu)(n) =

A theory of non-unitary unbounded similarity transformation operators is developed. To this end the class of J-unitary operators U is introduced. These operators are similar to unitary operators in their algebraic aspects but differ in their topological properties. It is shown how J-unitary operators are related to so-called J-biorthonormal systems and J-selfadjoint projections. Families {Uα} o...