Abstract
In this note we use Gelfand theory to show that the validity of a spectral mapping theorem for a given representation Φ:M → L(X) of a Banach function algebra M on a bounded open set in Cn implies the validity of one-sided spectral mapping theorems for all subspectra. In particular, a spectral mapping theorem for the Taylor spectrum yields at least a one-sided spectral mapping theorem for the essential Taylor spectrum. In our main exampleMis the multiplier algebra of a Banach space X of analytic functions and F is the canonical representation of M on X. In this case, we show that interpolating sequences for M, or suitably defined Berezin transforms, can sometimes be used to obtain the missing inclusion for the essential Taylor spectrum or its parts.
Mathematics Subject Classification. Primary 47A10, 47A13; secondary 47A60, 47B32.
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Reference
J. Agler, J.E. McCarthy, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, 44, AMS, Providence, RI, 2002.
M. Andersson, H. Carlsson, Estimates of solutions of the H p and BMOA corona problem, Math. Ann. 316 (2000), 83–102.
M. Andersson, S. Sandberg, The essential spectrum of holomorphic Toeplitz operators on H p spaces, Studia Math. 154 (2003), 223–231.
L. Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1958), 921-930
S. Costea, E. Sawyer, B.D. Wick, The Corona Theorem for the Drury-Arveson Hardy space and other holomorphic Besov Sobolev spaces on the unit ball in ℂn, Anal. PDE, to appear.
K.R. Davidson, R.G. Douglas, The generalized Berezin transform and commutator ideals, Pacific J. Math. 222 (2005), 29–56.
R.G. Douglas, J. Sarkar, A note on semi-Fredholm Hilbert modules, Operator Theory: Advances and Appl. 202, 143–150, Birkhäuser, Basel, 2010.
J. Eschmeier, Tensor products and elementary operators, J. reine angew. Math. 390 (1988), 47–66.
J. Eschmeier, M. Putinar, Spectral decompositions and analytic sheaves, LMS Mono-graph Series, Vol. 10, Clarendon Press, Oxford 1996.
C. Foiaş, W. Mlak, The extended spectrum of completely non-unitary contractions and the spectral mapping theorem, Studia Math. 26 (1966), 239–245.
L. Hormander, L 2-estimates and existence theorems for the ∂-operator, Acta Math. 113(1965), 89-152.
S.-Y. Li, Corona problems of several complex variables, Madison symposium of complex analysis, Contemporary Mathematics, vol. 137, Amer. Math. Soc., Providence, RI, 1991.
K.-C. Lin, The H p-corona theorem for the polydisc, Trans. Amer. Math. Soc. 341 (1994), 371-351.
Q. Lin, N. Salinas, Proper holomorphic maps and analytic Toeplitz n-tuples, Indiana Univ. Math. J. 39 (1990), 547–562.
K. Rudol, Spectral mapping theorems for analytic functional calculi, Operator Theory: Advances and Appl. 17 (1986), 331–340.
B. Sz.-Nagy, C. Foiaş, Harmonic analysis of operators on Hilbert spaces, NorthHolland, Amsterdam, 1970.
J.L. Taylor, The analytic functional calculus for several commuting operators, Acta Math. 125 (1970), 1–48.
T. Trent, On the Taylor spectrum of n-tuples of Toeplitz operators on the polydisc, multivariable operator theory workshop, Fields Institute, August 2009.
R. Zhao, K. Zhu, Theory of Bergman spaces on the unit ball of ℂn, Mém. Soc. Math. Fr. (NS), 115 (2008).
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Dedicated to Professor J. William Helton on the occasion of his 65th birthday
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Douglas, R.G., Eschmeier, J. (2012). Spectral Inclusion Theorems. In: Dym, H., de Oliveira, M., Putinar, M. (eds) Mathematical Methods in Systems, Optimization, and Control. Operator Theory: Advances and Applications, vol 222. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0411-0_10
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DOI: https://doi.org/10.1007/978-3-0348-0411-0_10
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