Abstract
We give necessary and sufficient conditions to prove a spectral theorem and a functional calculus for certain nonselfadjoint operators, H. Our method is non-perturbative: the conditions are given in terms of the resolvent (z-H)−1. We give an example of an operator satisfying these conditions. This operator is not a spectral operator of scalar type. Its spectral projections are unbounded operators defined on a common dense domainD.
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References
von Neumann, J.: Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, N.J., 1955.
Friedrichs, K. O.: On the Perturbation of Continuous Spectra, Comm. Pure Appl. Math. 1 (1948), 361–406.
Schwartz, J.: Some Non-Selfadjoint Operators, Comm. Pure Appl. Math. 13 (1960), 609–639.
Kuroda, S. T.: Perturbation of Continuous Spectra by Unbounded Operators I, J. Math. Soc. Japan 11 (1959) 247–262.
Kuroda, S. T.: Perturbation of Continuous Spectra by Unbounded Operators II, J. Math. Soc. Japan 12 (1960) 244–257.
Mochizuki, K.: On the Large Perturbation by a Class of Non-Selfadjoint Operators, J. Math. Soc. Japan 19 (1967), 123–158.
Goldstein, C.: Eigenfunction Expansions and Similarity for Certain Non-Selfadjoint Operators, Bull. Amer. Math. Soc. 75 (1969), 550–553.
Dunford, N. and Schwartz, J.: Linear Operators, Part III, Wiley-Interscience, New York, 1971.
Wermer, J.: Commuting Measures on Hilbert Space, Pacific J. Math. 4 (1954), 355–361
Kato, J.: Wave Operators and Similarity for some Non-Selfadjoint Operators, Math. Ann. 162 (1966), 258–279.
Angelescu, N., Marinescu, N. and Protopopescu, V., On the Spectral Decompositions of the Linear Transport Operator with Periodic Boundary Conditions, Lett. Math. Phys. 1 (1976), 329–333.
Harvey, B. N.: Spectral Operators with Critical Points, Amer. J. Math 96 (1974), 41–61.
Ramm, A. G.: Eigenfunction Exapnsions for some Non-Selfadjoint Operators and the Transport Equation, J. Math. Anal. Appl. 92 (1983), 564–580.
Sz.-Nagy, B. and Foias, C.: Harmonic Analysis of Operators on Hilbert Space, American Elsevier, New York, 1970.
Hoover, T. B.: Quasi-Similarity of Operators, Illinois J. Math. 16 (1972), 678–686.
Apostol, C.: Operators Quasi-Similar to a Normal Operator, Proc. Amer. Math. Soc. 53 (1975), 104–106.
Tzafriri, L.: Quasi-Similarity for Spectral Operators on Banach Spaces, Pacific J. Math 25 (1968), 197–217.
Feldzamen, A.: Semi-Similarity invariants for Spectral Operators on Hilbert Space, Trans. Amer. Math. Soc. 100 (1961), 277–323.
Akhiezer, N. I. and Glazman, I. M.: Theory of Linear Operators in Hilbert Space II, F. Ungar Pub. Co., New York, 1963.
Bowden, R. L., Sancaktar, S. and Zweifel, P. F.: Multigroup Neutron Transport I. Full Range, J. Math. Phys, 17 (1976), 76–81.
Dowson, H. R.: Spectral Theory of Linear Operators, Academic Press, New York, 1978.
Reed, M. and Simon, B.: Methods of Modern Mathematical Physics I, Academic Press, New York, 1972.
Yosida, K.: Functional Analysis, Springer-Verlag, New York, 1978.
Kolmogorov, A. N. and Fomin, S. V.: Introductory Real Analysis, Dover, New York, 1970.
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This research was supported in part by Department of Energy Grant No. DE-AS05-80ER10711 and National Science Foundation Grant No. DMA-8312451.
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Burnap, C., Zweifel, P.F. A note on the spectral theorem. Integr equ oper theory 9, 305–324 (1986). https://doi.org/10.1007/BF01199348
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DOI: https://doi.org/10.1007/BF01199348