Norm estimates for functions of non-selfadjoint operators nonregular on the convex hull of the spectrum

• Autorzy: Gil’ M.

Identyfikatory

DOI

10.1515/dema-2017-0026

• Języki publikacji: EN

Abstrakty

EN

We consider a bounded linear operator A in a Hilbert space with a Hilbert-Schmidt Hermitian component (A−A*)/2i. A sharp norm estimate is established for functions of A nonregular on the convex hull of the spectrum. The logarithm, fractional powers and meromorphic functions of operators are examples of such functions. Our results are based on the existence of a sequence An(n = 1, 2,...) of finite dimensional operators strongly converging to A, whose spectra belongs to the spectrum of A. Besides, it is shown that the resolvents and holomorphic functions of An strongly converge to the resolvent and corresponding function of A.

Słowa kluczowe

EN

functions of non-selfadjoint operators; logarithm; fractional power; meromorphic function

PL

operator samosprzężony; operator hermitowski; logarytm; funkcja meromorficzna

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2017
• Tom: Vol. 50, nr 1
• Strony: 267--277
• Opis fizyczny: Bibliogr. 23 poz.

Twórcy

autor

Gil’ M.

• Department of Mathematics, Ben Gurion University of the Negev, P.0. Box 653, Beer-Sheva 84105, Israel

Bibliografia

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• [6] Boyadzhiev K. N., Logarithms and imaginary powers of operators on Hilbert spaces, Collect. Math., 1994, 45, 287-300
• [7] Chiumiento E., On normal operator logarithms, Linear Algebra Appl., 2013, 439, 455-462
• [8] Conway J. B., Morrel B. B., Roots and logarithms of bounded operators on Hilbert space, J. Funct. Anal., 1987, 70(1), 171-193
• [9] Haase M., Spectral properties of operator logarithms, Math. Z., 2003, 245, 761-779
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• [11] Schmoeger C., On logarithms of linear operators on Hilbert spaces, Demonstr. Math., 2002, 35(2), 375-384
• [12] Gil’ M. I., Matrix functions nonregular on the convex hull of the spectrum, Linear Multilinear Algebra, 2012, 60(4), 465-473
• [13] Gohberg I. C., Goldberg S., Kaashoek M. A., Classes of linear operators, 2, Birkhäuser Verlag, Basel, 1993
• [14] Brodskii M. S., Triangular and Jordan representations of linear operators, Transl. Math. Mongr., 32, Amer. Math. Soc., Providence, R. I., 1971
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• [16] Radjavi H., Rosenthal P., Invariant subspaces, Springer-Verlag, Berlin, 1973
• [17] Gil’ M. I., Bounds for determinants of linear operators and their applications, CRC Press, Taylor & Francis Group, London, 2017
• [18] Brodskii V. M., Gohberg I. C., Krein M. G., General theorems on triangular representations of linear operators and multiplicative representations of their characteristic functions, Funct. Anal. Appl., 1969, 3(4), 255-276
• [19] Brodskii M. S., Triangular representation of some operators with completely continuous imaginary parts, Dokl. Akad. Nauk SSSR, 1960, 133, 1271-1274 (in Russian). English translation: Soviet Math. Dokl., 1960, 1, 952-955
• [20] Gohberg I. C., Krein M. G., Introduction to the theory of linear nonselfadjoint operators, Trans. Mathem. Monographs, 18, Amer. Math. Soc., R. I., 1969
• [21] Kato T., Perturbation theory for linear operators, Springer-Verlag, Berlin 1980
• [22] Bhatia R., Rosenthal P., How and why to solve the matrix equation AX−XB=Y, Bull. London Math. Soc., 1997, 29, 1-21
• [23] Gil’ M. I., Resolvents of operators on tensor products of Euclidean spaces, Linear Multilinear Algebra, 2015, 64(4), 699-716

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).