Invariant Subspaces for Closed ∗-Representations of ∗-Algebras

O*‐algebras in Standard System

“…In this section we continue the study of generalized vectors for 0*-algebras studied in [12]. We begin with some of definitions and the basic properties concerning 0*-algebras, and refer to the papers [3,6,7,12,141 and the SCHMUDGEN book [19] in more details.…”

“…Let m be an A-invariant subspace of 9 and &z' r m the set of all restrictions X r m of X E A to m. Then A r m is an O*-algebra on m. If A is self-adjoint, then A r m is essentially self-adjoint if and only if E m 9 equals the closure ='A of m with respect to the graph topology t,, where Em 3 Proj fi [14]. An element 5 of 9 is said to be a self-adjoint vector for 4 if 4 r A 5 is essentially self-adjoint [6]. A closed 0'-algebra A is said to be standard if X * = 3 for each X E A.…”

Cyclic Generalized Vectors for Algebras of Unbounded Operators

“…In this section we state some definitions and basic properties concerning O*-algebras [3,6,11,13,15,18,19].…”

“…of the seminorms : [15] . An element f of ^ is said to be a self-adjoint vector for ^ if M\ ~ M% is essentially self-adjoint [6] .…”