Vol. 42, No. 2, 1972

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Invariant subspaces and operators of class (S)

Norberto Salinas

Vol. 42 (1972), No. 2, 497–513

Let be an infinite dimensional separable complex Hilbert space, and let () denote the algebra of all (bounded linear) operators on . This paper is concerned with a specific class of two-by-two operator matrices acting in the usual fashion on p ⊕ℋ. An operator in (ℋ⊕ℋ) will be said to be of class (S) if it can be represented as a two by two operator matrix of the form

[       ]
A    V
− V ∗ 0

where V is a unilateral shift of infinite multiplicity on and A is an arbitrary operator in ().

In the present paper it is shown that the study of the operators of class (S) arises naturally in connection with the invariant subspace problem. In particular, the question of whether an operator of class (S) has a nontrivial invariant subspace is raised, and some significant results are obtained toward the solution of this problem.

Mathematical Subject Classification 2000
Primary: 47A15
Received: 19 January 1971
Published: 1 August 1972
Norberto Salinas