Vol. 30, No. 3, 1969

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Note on some spectral inequalities of C. R. Putnam

Sterling K. Berberian

Vol. 30 (1969), No. 3, 573–575
Abstract

It is shown that if A is any operator in Hilbert space and λ = rei𝜃 is in the approximate point spectrum of A, then

minA ∗A ≦ (max J𝜃)2

and

|r− max J𝜃| ≦ |(max J𝜃)2 − min A∗A|1∕2,

where

            −i𝜃    ∗i𝜃
J𝜃 = (1∕2)(Ae   + A e  ).

Several corollaries are deduced for arbitrary operators, generalizing results of C. R. Putnam on semi-normal operators.

Mathematical Subject Classification
Primary: 47.30
Milestones
Received: 26 November 1968
Published: 1 September 1969
Authors
Sterling K. Berberian