Some Invariant subspaces for A-contractions and applications

Abstract

Let H be a complex Hilbert space and B(H) the Banach algebra of all bounded linear operators on H. If T ∈ B(H) then T ∗ stands for the adjoint operator of T, while R(T) and N (T) denote the range and the null-space of T, respectively. A contraction onH is an operator T ∈ B(H) satisfying T ∗T ≤ I, where I = IH is the identity operator. If T ∗T < I then T is called a proper contraction. The class of contractions is one of the most studied and well-understood class of operators (see for instance [2], [3], [6], [11]) and the investigations concerning different other classes in B(H) have a starting point the theory of contractions. We refer below to a class of operators which generalize the contractions. Let A ∈ B(H) be a positive operator, A 6 = 0. An operator T ∈ B(H) satisfying the inequality T ∗AT ≤ A(1.1) is called an A-contraction on H. If the equality in (1.1) one occurs then T is called an A-isometry on H. Such operators appear in different contexts in [1], [2], [3], [5], [7]–[10], [11], and other papers. By contrast to the class of contractions (that is, of I-contractions), the class of A-contractions is not