Holomorphic kernels and commuting operators
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- by Ameer Athavale
- Trans. Amer. Math. Soc. 304 (1987), 101-110
- DOI: https://doi.org/10.1090/S0002-9947-1987-0906808-6
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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
- Jim Agler, The Arveson extension theorem and coanalytic models, Integral Equations Operator Theory 5 (1982), no. 5, 608–631. MR 697007, DOI 10.1007/BF01694057
- Jim Agler, Hypercontractions and subnormality, J. Operator Theory 13 (1985), no. 2, 203–217. MR 775993
- N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404. MR 51437, DOI 10.1090/S0002-9947-1950-0051437-7
- William B. Arveson, Subalgebras of $C^{\ast }$-algebras, Acta Math. 123 (1969), 141–224. MR 253059, DOI 10.1007/BF02392388
- Joseph Bram, Subnormal operators, Duke Math. J. 22 (1955), 75–94. MR 68129
- T. H. Hildebrandt and I. J. Schoenberg, On linear functional operations and the moment problem for a finite interval in one or several dimensions, Ann. of Math. (2) 34 (1933), no. 2, 317–328. MR 1503109, DOI 10.2307/1968205
- Takasi Itô, On the commutative family of subnormal operators, J. Fac. Sci. Hokkaido Univ. Ser. I 14 (1958), 1–15. MR 0107177
- Arthur Lubin, Weighted shifts and commuting normal extension, J. Austral. Math. Soc. Ser. A 27 (1979), no. 1, 17–26. MR 524154
- Raghavan Narasimhan, Several complex variables, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1971. MR 0342725 J. R. Ringrose, Compact non-self-adjoint operators, Van Nostrand Reinhold, London, 1971.
- F. H. Szafraniec, Dilations on involution semigroups, Proc. Amer. Math. Soc. 66 (1977), no. 1, 30–32. MR 473873, DOI 10.1090/S0002-9939-1977-0473873-1 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.
- Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190
Bibliographic Information
- © Copyright 1987 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 304 (1987), 101-110
- MSC: Primary 47B20; Secondary 47A20
- DOI: https://doi.org/10.1090/S0002-9947-1987-0906808-6
- MathSciNet review: 906808