Bounded and Unbounded Linear Operators

Abstract

Let H be a complex inner product space and let L(H) be the set of all bounded linear operators on H. In this chapter we present some basic facts about the set L(H) as well as about the set of unbounded operators on H. We also present some classes of operators whose structure is better understood, among which we mention the class of hermitian, unitary and normal operators, selfadjoint and essentially selfadjoint operators, and the class of unbounded normal operators. Several examples illustrate the exposed results. Also, since there has recently been a growing interest in new classes of unbounded operators, we also present some classes of unbounded operators related to bounded normal operators, namely the classes of quasinorma1 operators, subnormal operators, hyponormal operators, operators of class (N), W–N operators, θ operators, etc. For more detail on the classes of bounded operators we refer to the author’s book (1981).