Spectra and measure inequalities

Author:
C. R. Putnam

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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let *T* be a bounded operator on a Hilbert space and let . Then the operators , and 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 be separable and suppose that 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 . If *T* is the adjoint of the simple unilateral shift, then equality holds with .

**[1]**James E. Brennan,*Invariant subspaces and rational approximation*, J. Functional Analysis**7**(1971), 285–310. MR**0423059****[2]**Kevin F. Clancey and Bernard B. Morrel,*The essential spectrum of some Toeplitz operators*, Proc. Amer. Math. Soc.**44**(1974), 129–134. MR**0341162**, https://doi.org/10.1090/S0002-9939-1974-0341162-9**[3]**J. Dixmier and C. Foiaş,*Sur le spectre ponctuel d’un opérateur*, Hilbert space operators and operator algebras (Proc. Internat. Conf., Tihany, 1970) North-Holland, Amsterdam, 1972, pp. 127–133. Colloq. Math. Soc. János Bolyai, No. 5 (French). MR**0365175****[4]**Theodore W. Gamelin,*Uniform algebras*, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. MR**0410387****[5]**John Garnett,*Analytic capacity and measure*, Lecture Notes in Mathematics, Vol. 297, Springer-Verlag, Berlin-New York, 1972. MR**0454006****[6]**B. E. Johnson,*Continuity of linear operators commuting with continuous linear operators*, Trans. Amer. Math. Soc.**128**(1967), 88–102. MR**0213894**, https://doi.org/10.1090/S0002-9947-1967-0213894-5**[7]**L. N. Nikol′skaja,*Structure of the point spectrum of a linear operator*, Mat. Zametki**15**(1974), 149–158 (Russian). MR**0346554****[8]**C. R. Putnam,*An inequality for the area of hyponormal spectra*, Math. Z.**116**(1970), 323–330. MR**0270193**, https://doi.org/10.1007/BF01111839**[9]**C. R. Putnam,*Ranges of normal and subnormal operators*, Michigan Math. J.**18**(1971), 33–36. MR**0276810****[10]**C. R. Putnam,*Resolvent vectors, invariant subspaces, and sets of zero capacity*, Math. Ann.**205**(1973), 165–171. MR**0326428**, https://doi.org/10.1007/BF01350843**[11]**C. R. Putnam,*Hyponormal contractions and strong power convergence*, Pacific J. Math.**57**(1975), no. 2, 531–538. MR**0380493****[12]**Walter Rudin,*Real and complex analysis*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210528****[13]**Lawrence Zalcmann,*Analytic capacity and rational approximation*, Lecture Notes in Mathematics, No. 50, Springer-Verlag, Berlin-New York, 1968. MR**0227434**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
47A30

Retrieve articles in all journals with MSC: 47A30

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1977-0487511-X

Keywords:
Hilbert space,
spectra of operators,
hyponormal operators,
analytic capacity,
continuous analytic capacity,
planar measure

Article copyright:
© Copyright 1977
American Mathematical Society