Vol. 30, No. 2, 1969

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ISSN: 0030-8730
Semi-groups of scalar type operators in Banach spaces

T. V. Panchapagesan

Vol. 30 (1969), No. 2, 489–517

This paper deals with the spectral representation theorems of semi-groups of scalar type operators in Banach spaces. These results generalize the corresponding ones on semi-groups of hermitian, normal and unitary operators in Hilbert spaces. In the beginning sections we study some interesting properties of a W(∥⋅∥)-algebra-which generalizes the notion of an abelian von Neumann algebra to Banach spaces-and unbounded spectral operators arising out of E()-unbounded measurable functions where E() is a resolution of the identity. These results are applied later to prove the spectral representation theorems on semi-groups of scalar type operators. The last theorem of this paper gives an extension of Stone’s theorem on strongly continuous one parameter group of unitary operators to arbitrary Banach spaces.

Mathematical Subject Classification 2000
Primary: 47D05
Secondary: 47B40
Received: 1 March 1968
Published: 1 August 1969
T. V. Panchapagesan