Best approximation of a normal operator in the Schatten $p$-norm
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- by Richard Bouldin PDF
- Proc. Amer. Math. Soc. 80 (1980), 277-282 Request permission
Abstract:
Let A be a fixed normal operator and let $\mathfrak {N}(\Lambda )$ denote the normal operators with spectrum contained in $\Lambda$. Provided there is some N in $\mathfrak {N}(\Lambda )$ such that $A - N$ belongs to the Schatten class ${c_p},p \geqslant 2$, the main result of this paper obtains a best approximation for A from $\mathfrak {N}(\Lambda )$ with respect to the Schatten p-norm. A necessary and sufficient condition is given for A to have a unique best approximation in that case.References
-
J. G. Aiken, H. B. Jonassen and H. S. Aldrich, Löwdin orthonormalization as a minimum energy perturbation, J. Chem. Phys. 62 (1975), 2745-2746.
J. G. Aiken, J. A. Erdös and J. A. Goldstein, Unitary approximation of positive operators (preprint).
- S. K. Berberian, The Weyl spectrum of an operator, Indiana Univ. Math. J. 20 (1970/71), 529–544. MR 279623, DOI 10.1512/iumj.1970.20.20044 M. S. Birman and M. Z. Solomyak, Stieltjes double-integral operators, Spectral Theory and Wave Processes (Topics in Mathematical Physics, vol. 1), Consultants Bureau, New York, 1976.
- Richard Bouldin, Essential spectrum for a Hilbert space operator, Trans. Amer. Math. Soc. 163 (1972), 437–445. MR 284837, DOI 10.1090/S0002-9947-1972-0284837-5
- B. C. Carlson and Joseph M. Keller, Orthogonalization procedures and the localization of Wannier functions, Phys. Rev. (2) 105 (1957), 102–103. MR 90410, DOI 10.1103/PhysRev.105.102
- A. J. Coleman, Structure of fermion density matrices, Rev. Modern Phys. 35 (1963), 668–689. MR 0155637, DOI 10.1103/RevModPhys.35.668
- Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, Interscience Publishers John Wiley & Sons, New York-London, 1963. With the assistance of William G. Bade and Robert G. Bartle. MR 0188745
- Ky Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 760–766. MR 45952, DOI 10.1073/pnas.37.11.760
- I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR 0246142, DOI 10.1090/mmono/018
- Karl Gustafson, Necessary and sufficient conditions for Weyl’s theorem, Michigan Math. J. 19 (1972), 71–81. MR 295112
- P. R. Halmos, Spectral approximants of normal operators, Proc. Edinburgh Math. Soc. (2) 19 (1974/75), 51–58. MR 344935, DOI 10.1017/S0013091500015364
- Frederick A. Valentine, Convex sets, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Toronto-London, 1964. MR 0170264
- Hermann Weyl, Inequalities between the two kinds of eigenvalues of a linear transformation, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 408–411. MR 30693, DOI 10.1073/pnas.35.7.408
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 80 (1980), 277-282
- MSC: Primary 47B20
- DOI: https://doi.org/10.1090/S0002-9939-1980-0577759-3
- MathSciNet review: 577759