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Let ℋ be an infinite
dimensional separable complex Hilbert space, and let ℒ(ℋ) denote the algebra of all
(bounded linear) operators on ℋ. This paper is concerned with a specific class of
two-by-two operator matrices acting in the usual fashion on ℋp ⊕ℋ. An operator in
ℒ(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](a170x.png)
where V is a unilateral shift of infinite multiplicity on ℋ and A is an arbitrary
operator in ℒ(ℋ).
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.
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