Vol. 22, No. 3, 1967

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Unitary operators in Banach spaces

T. V. Panchapagesan

Vol. 22 (1967), No. 3, 465–475

The notion of hermitian operators in Hilbert space has been extended to Banach spaces by Lumer and Vidav. Recently, Berkson has shown that a scalar type operator S in a Banach space X can be decomposed into S = R + iJ where (i) R and J commute and (ii) RmJn(m,n = 0,1,2,) are hermitian in some equivalent norm on X. The converse is also valid if the Banach space is reflexive. Thus we see that the scalar type operators in a Banach space play a role analogous to the normal operators in a Hilbert space.

In this paper, the well-known Hilbert space notion of unitary operators is suitably extended to operators in Banach spaces and a polar decomposition is obtained for a scalar type operator. It is further shown that this polar decomposition is unique and characterises scalar type operators in reflexive Banach spaces. Finally, an extension of stone’s theorem on one-parameter group of unitary operators in Hilbert spaces is obtained (under suitable conditions) for reflexive Banach spaces.

Mathematical Subject Classification
Primary: 47.40
Received: 28 April 1964
Revised: 4 October 1966
Published: 1 September 1967
T. V. Panchapagesan