M Class Q Composition Operators (Report‪)‬

[section]1. Introduction and preliminaries Let H be a complex Hilbert space and B(H) the algebra of all bounded linear operators on H. An operator T [member of] B(H) is M Paranormal if for a fixed real number M > 0, T satisfies [[parallel]Tx[parallel].sup.2] [less than or equal to] M[parallel][T.sup.2]x[parallel][parallel]x[parallel] for every x [member of] H, normaloid if r(T) = [parallel]T[parallel], where r(T) denotes the spectral radius of T and class Q, T [member of] Q, if [T.sup.*2][T.sup.2] - 2[T.sup.*]T + 1 [greater than or equal to] 0. Equivalently T [member of] Q if [[parallel]Tx[parallel].sup.2] [less than or equal to] 1/2 [[[parallel]T.sup.2]x[parallel].sup.2] + [[parallel]x[parallel].sup.2]] for every x [member of] H. Class Q operators are inroduced and studied by B. P Duggal et al. [6] and it is well known that every class Q operator is not necessarily normaloid and every paranormal operator is a normaloid of class Q, that is P [subset not equal to] Q [intersection] N, where P and N denotes for class of normaloid and paranormal operators. A contraction is an operator T such that [parallel]T[parallel] [less than or equal to] 1 (i.e [parallel]Tx[parallel] [less than or equal to] [parallel]x[parallel] for every x [member of] H, equivalently [T.sup.*]T [less than or equal to] 1). By a subspace M of H, we mean a closed linear manifold of H. A subspace M is invariant for T if T(M) [subset not equal to] M and a part of an operator is a restriction of it to an invariant subspace.