Vol. 62, No. 2, 1976

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On partial isometries with no isometric part

James Guyker

Vol. 62 (1976), No. 2, 419–433

P. R. Halmos and L. J. Wallen characterized partial isometries all of whose positive integral powers are partial isometries on Hilbert space as unique direct sums of unitary operators, pure isometries, pure co-isometries and truncated shifts, with each type of summand occurring at most once. In the present paper, this structure is extended to partial isometries T with no isometric part whose first N + 1 powers are partial isometries. Moreover, if T = Σ Tj V , it is shown that every projection P commuting with T is of the form P = Σ Pj Q where Pj and Q are projections commuting with Tj and V respectively. The canonical model of L. de Branges and J. Rovnyak is used to explicitly describe the structure of the reducing subspaces of T in terms of the characteristic operator-function of T, from which this result follows. A direct proof is also obtained using a general reducing subspace structure theorem for arbitrary contractions with no isometric part.

Mathematical Subject Classification 2000
Primary: 47A45
Received: 12 March 1975
Revised: 20 January 1976
Published: 1 February 1976
James Guyker