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 | References | Similar Articles | Additional Information

Abstract: Necessary and sufficient conditions in terms of operator polynomials are obtained for an -tuple of commuting bounded linear operators on a separable Hilbert space to extend to an -tuple of operators on some Hilbert space , where each is realized as a -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 to have a commuting normal extension.

**[1]**Jim Agler,*The Arveson extension theorem and coanalytic models*, Integral Equations Operator Theory**5**(1982), no. 5, 608–631. MR**697007**, 10.1007/BF01694057**[2]**Jim Agler,*Hypercontractions and subnormality*, J. Operator Theory**13**(1985), no. 2, 203–217. MR**775993****[3]**N. Aronszajn,*Theory of reproducing kernels*, Trans. Amer. Math. Soc.**68**(1950), 337–404. MR**0051437**, 10.1090/S0002-9947-1950-0051437-7**[4]**William B. Arveson,*Subalgebras of 𝐶*-algebras*, Acta Math.**123**(1969), 141–224. MR**0253059****[5]**Joseph Bram,*Subnormal operators*, Duke Math. J.**22**(1955), 75–94. MR**0068129****[6]**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**, 10.2307/1968205**[7]**Takasi Itô,*On the commutative family of subnormal operators*, J. Fac. Sci. Hokkaido Univ. Ser. I**14**(1958), 1–15. MR**0107177****[8]**Arthur Lubin,*Weighted shifts and commuting normal extension*, J. Austral. Math. Soc. Ser. A**27**(1979), no. 1, 17–26. MR**524154****[9]**Raghavan Narasimhan,*Several complex variables*, The University of Chicago Press, Chicago, Ill.-London, 1971. Chicago Lectures in Mathematics. MR**0342725****[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), no. 1, 30–32. MR**0473873**, 10.1090/S0002-9939-1977-0473873-1**[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]**Béla Sz.-Nagy and Ciprian Foiaș,*Harmonic analysis of operators on Hilbert space*, Translated from the French and revised, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. MR**0275190**

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DOI:
https://doi.org/10.1090/S0002-9947-1987-0906808-6

Keywords:
Positive definite,
regular analytic model atom,
kernel,
tensor product,
commuting normal extension,
-dilation

Article copyright:
© Copyright 1987
American Mathematical Society