Perturbation determinants on Banach spaces and operator differentiability for Hirsch functional calculus. (English) Zbl 1498.47036
Summary: We consider a perturbation determinant for pairs of nonpositive (in the sense of H. Komatsu [J. Math. Soc. Japan 21, 205–220, 221–228 (1969; Zbl 0181.41003)]) operators on Banach space with nuclear difference and prove the formula for the logarithmic derivative of this determinant. To this end, the Fréchet differentiability of operator monotonic (negative complete Bernstein) functions of negative and nonpositive operators on Banach spaces is investigated. The results may be regarded as a contribution to the Hirsch functional calculus.
MSC:
| 47A60 | Functional calculus for linear operators |
| 47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |
| 47L20 | Operator ideals |
Keywords:
perturbation determinant; nonpositive operator; Hirsch functional calculus; Bernstein function; operator monotonic function; operator differentiabilityCitations:
Zbl 0181.41003References:
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