Spectra and measure inequalities
HTML articles powered by AMS MathViewer
- by C. R. Putnam PDF
- Trans. Amer. Math. Soc. 231 (1977), 519-529 Request permission
Abstract:
Let T be a bounded operator on a Hilbert space $\mathfrak {H}$ and let ${T_z} = T - zI$. Then the operators ${T_z}T_z^\ast ,{T_z}{T_t}{({T_z}{T_t})^\ast }$, and ${T_z}{T_t}{T_s}{({T_z}{T_t}{T_s})^\ast }$ are nonnegative for all complex numbers z, t, and s. We shall obtain some norm estimates for nonnegative lower bounds of these operators, when z, t, and s are restricted to certain sets, in terms of certain capacities or area measures involving the spectrum and point spectrum of T. A typical such estimate is the following special case of Theorem 4 below: Let $\mathfrak {H}$ be separable and suppose that ${T_z}{T_t}{({T_z}{T_t})^\ast } \geqslant D \geqslant 0$ for all z and t not belonging to the closure of the interior of the point spectrum of T. In addition, suppose that the boundary of the interior of the point spectrum of T has Lebesgue planar measure 0. Then ${\left \| D \right \|^{1/2}} \leqslant {\pi ^{ - 1}}\;{\text {meas}_2}\;({\sigma _p}(T))$. If T is the adjoint of the simple unilateral shift, then equality holds with $D = I - {T^\ast }T$.References
- James E. Brennan, Invariant subspaces and rational approximation, J. Functional Analysis 7 (1971), 285–310. MR 0423059, DOI 10.1016/0022-1236(71)90036-x
- Kevin F. Clancey and Bernard B. Morrel, The essential spectrum of some Toeplitz operators, Proc. Amer. Math. Soc. 44 (1974), 129–134. MR 341162, DOI 10.1090/S0002-9939-1974-0341162-9
- J. Dixmier and C. Foiaş, Sur le spectre ponctuel d’un opérateur, Hilbert space operators and operator algebras (Proc. Internat. Conf., Tihany, 1970) Colloq. Math. Soc. János Bolyai, vol. 5, North-Holland, Amsterdam, 1972, pp. 127–133 (French). MR 0365175
- Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
- John Garnett, Analytic capacity and measure, Lecture Notes in Mathematics, Vol. 297, Springer-Verlag, Berlin-New York, 1972. MR 0454006
- B. E. Johnson, Continuity of linear operators commuting with continuous linear operators, Trans. Amer. Math. Soc. 128 (1967), 88–102. MR 213894, DOI 10.1090/S0002-9947-1967-0213894-5
- L. N. Nikol′skaja, Structure of the point spectrum of a linear operator, Mat. Zametki 15 (1974), 149–158 (Russian). MR 346554
- C. R. Putnam, An inequality for the area of hyponormal spectra, Math. Z. 116 (1970), 323–330. MR 270193, DOI 10.1007/BF01111839
- C. R. Putnam, Ranges of normal and subnormal operators, Michigan Math. J. 18 (1971), 33–36. MR 276810
- C. R. Putnam, Resolvent vectors, invariant subspaces, and sets of zero capacity, Math. Ann. 205 (1973), 165–171. MR 326428, DOI 10.1007/BF01350843
- C. R. Putnam, Hyponormal contractions and strong power convergence, Pacific J. Math. 57 (1975), no. 2, 531–538. MR 380493
- Walter Rudin, Real and complex analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210528
- Lawrence Zalcmann, Analytic capacity and rational approximation, Lecture Notes in Mathematics, No. 50, Springer-Verlag, Berlin-New York, 1968. MR 0227434
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 231 (1977), 519-529
- MSC: Primary 47A30
- DOI: https://doi.org/10.1090/S0002-9947-1977-0487511-X
- MathSciNet review: 0487511