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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Holomorphic kernels and commuting operators

Author: Ameer Athavale
Journal: Trans. Amer. Math. Soc. 304 (1987), 101-110
MSC: Primary 47B20; Secondary 47A20
MathSciNet review: 906808
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Abstract: Necessary and sufficient conditions in terms of operator polynomials are obtained for an $ m$-tuple $ T = ({T_1}, \ldots ,{T_m})$ of commuting bounded linear operators on a separable Hilbert space $ \mathcal{H}$ to extend to an $ \dot m$-tuple $ S = ({S_1}, \ldots ,{S_m})$ of operators on some Hilbert space $ \mathcal{K}$, where each $ {S_i}$ is realized as a $ {\ast}$-representation of the adjoint of a multiplication operator on the tensor product of a special type of functional Hilbert spaces. Also, necessary and sufficient conditions in terms of operator polynomials are obtained for $ T$ to have a commuting normal extension.

References [Enhancements On Off] (What's this?)

  • [1] J. Agler, The Arveson extension theorem and coanalytic models, Integral Equations Operator Theory 5 (1982), 608-631. MR 697007 (84g:47011)
  • [2] -, Hypercontractions and subnormality, J. Operator Theory 13 (1985), 203-217. MR 775993 (86i:47028)
  • [3] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337-404. MR 0051437 (14:479c)
  • [4] W. B. Arveson, Subalgebras of $ {C^{\ast}}$, Acta Math. 123 (1969), 141-224. MR 0253059 (40:6274)
  • [5] J. Bram, Subnormal operators, Duke Math. J. 22 (1955), 75-94. MR 0068129 (16:835a)
  • [6] T. N. Hildebrandt and I. J. Schoenberg, On linear functional operators and the moment problem for a finite interval in one or several dimensions, Ann. of Math. 34 (1933), 317-328. MR 1503109
  • [7] T. Ito, On the commutative family of subnormal operators, J. Fac. Sci. Hokkaido Univ. 14 (1958), 1-15. MR 0107177 (21:5902)
  • [8] A. Lubin, Weighted shifts and commuting normal extension, J. Austral. Math. Soc. 27 (1979), 17-26. MR 524154 (80j:47034)
  • [9] R. Narasimhan, Several complex variables, Univ. of Chicago Press, Chicago and London, 1971. MR 0342725 (49:7470)
  • [10] J. R. Ringrose, Compact non-self-adjoint operators, Van Nostrand Reinhold, London, 1971.
  • [11] F. H. Szafraniec, Dilations on involution semigroups, Proc. Amer. Math. Soc. 66 (1977), 30-32. MR 0473873 (57:13532)
  • [12] B. Sz.-Nagy, Extensions of linear transformations in Hilbert space which extend beyond this space, Appendix to F. Riesz and B. Sz.-Nagy, Functional Analysis, Ungar, New York, 1960.
  • [13] B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, North-Holland, Amsterdam, 1970. MR 0275190 (43:947)

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Additional Information

PII: S 0002-9947(1987)0906808-6
Keywords: Positive definite, regular analytic model atom, kernel, tensor product, commuting normal extension, $ {\ast}$ -dilation
Article copyright: © Copyright 1987 American Mathematical Society