Vol. 36, No. 3, 1971

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Operators that commute with a unilateral shift on an invariant subspace

Lavon Barry Page

Vol. 36 (1971), No. 3, 787–794

A co-isometry on a Hilbert space is a bounded operator having an isometric adjoint. If V is a co-isometry on and is an invariant subspace for V , then every bounded operator on that commutes with V on can be extended to an operator on that commutes with V , and the extension can be made without increasing the norm of the operator. This paper is concerned with unilateral shifts. The questions asked are these: (1) Do shifts enjoy the above property shared by co-isometries and self-adjoint operators? (The answer to this question is “rarely”.) (2) Why not? (3) If S is a shift, is an invariant subspace for S, S0 is the restriction of S to , and T is a bounded operator on satisfying TS0 = S0T, how tame do T and have to be in order that T can be extended (without increasing the norm) to an operator in the commutant of S? Extension is possible in a large number of cases.

Mathematical Subject Classification
Primary: 47.40
Received: 6 April 1970
Published: 1 March 1971
Lavon Barry Page