Garcia, Stephan Ramon

Scholarship@Claremont

Claremont CollegesScholarship @ ClaremontPomona Faculty Publications and Research Pomona Faculty Scholarship7-14-2015Review: Nevanlinna-Pick spaces with hyponormalmultiplication operatorsStephan Ramon GarciaPomona CollegeThis Review is brought to you for free and open access by the Pomona Faculty Scholarship at Scholarship @ Claremont. It has been accepted forinclusion in Pomona Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, pleasecontact scholarship@cuc.claremont.edu.Recommended CitationMR3336615 Hartz, M., Nevanlinna-Pick spaces with hyponormal multiplication operators, Proc. Amer. Math. Soc. 143 (2015), no. 7,2905-2912. (Reviewer: Stephan R. Garcia)Previous Up NextCitationsFrom References: 0From Reviews: 0MR3336615 (Review) 46E22 47B20 47B32Hartz, Michael (3-WTRL-PM)Nevanlinna-Pick spaces with hyponormal multiplication operators. (Englishsummary)Proc. Amer. Math. Soc. 143 (2015), no. 7, 2905–2912.The main result of this paper (Theorem 1.1) shows that the classical Hardy space H2is essentially the only complete Nevanlinna-Pick space whose multiplication operatorsare all hyponormal. To be more specific, if H is an irreducible complete Nevanlinna-Pick space on a set X with kernel K such that all multiplication operators on H arehyponormal, then either:1. X is a singleton and H= C.2. There is a set of uniqueness A ⊂ D for H2, a bijection j:X → A, and a nowherevanishing function δ:X → C such thatK(λ, µ) = δ(λ)δ(µ)k(j(λ), j(µ)).Here k(z, w) denotes the Szego˝ kernel on H2.In particular, the map f 7→ δ(f ◦ j) is a unitary operator. Continuity and analyticity arealso respected: if X is endowed with a topology such that K is separately continuous onX ×X, then j is also continuous. If X ⊂ Cn and K is holomorphic in the first variable,then j is holomorphic.An interesting consequence of the main theorem is Corollary 1.3, which asserts that ifn≥ 3 and U ⊂ Rn is an open set, then there is no irreducible complete Nevanlinna-Pickspace on U which consists of continuous functions and whose multiplication operatorsare all hyponormal. The proof appeals to Brouwer’s invariance of domain theorem.Stephan R. GarciaReferences1. Jim Agler and John E. McCarthy, Complete Nevanlinna-Pick kernels, J. Funct.Anal. 175 (2000), no. 1, 111–124, DOI 10.1006/jfan.2000.3599. MR1774853(2001h:47019) MR17748532. Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces,Graduate Studies in Mathematics, vol. 44, American Mathematical Society, Provi-dence, RI, 2002. MR18822593. Ca˘lin-Grigore Ambrozie and Dan Timotin, On an intertwining lifting theoremfor certain reproducing kernel Hilbert spaces, Integral Equations Operator Theory42 (2002), no. 4, 373–384, DOI 10.1007/BF01270919. MR1885440 (2002m:47012)MR18854404. William Arveson, Subalgebras of C∗-algebras. III. Multivariable operator theory,Acta Math. 181 (1998), no. 2, 159–228, DOI 10.1007/BF02392585. MR1668582(2000e:47013) MR16685825. Joseph A. Ball, Tavan T. Trent, and Victor Vinnikov, Interpolation and commutantlifting for multipliers on reproducing kernel Hilbert spaces, Operator theory andanalysis (Amsterdam, 1997), Oper. Theory Adv. Appl., vol. 122, Birkha¨user, Basel,2001, pp. 89–138. MR1846055 (2002f:47028) MR18460556. L. E. J. Brouwer, Beweis der invarianz des n-dimensionalen gebiets (German), Math.Ann. 71 (1911), no. 3, 305–313, DOI 10.1007/BF01456846. MR1511658 MR15116587. John B. Garnett, Bounded analytic functions, 1st ed., Graduate Texts in Mathe-matics, vol. 236, Springer, New York, 2007. MR2261424 (2007e:30049) MR22614248. Allen L. Shields, Weighted shift operators and analytic function theory, Topicsin operator theory, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. Math.Surveys, No. 13. MR0361899 (50 #14341) MR0361899Note: This list reflects references listed in the original paper as accurately aspossible with no attempt to correct errors.c© Copyright American Mathematical Society 2016

Review: Nevanlinna-Pick spaces with hyponormal multiplication operators

Keck Graduate Institute

Claremont Colleges
Scholarship @ Claremont
Pomona Faculty Publications and Research Pomona Faculty Scholarship
7-14-2015
Review: Nevanlinna-Pick spaces with hyponormal
multiplication operators
Stephan Ramon Garcia
Pomona College
This Review is brought to you for free and open access by the Pomona Faculty Scholarship at Scholarship @ Claremont. It has been accepted for
inclusion in Pomona Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, please
contact scholarship@cuc.claremont.edu.
Recommended Citation
MR3336615 Hartz, M., Nevanlinna-Pick spaces with hyponormal multiplication operators, Proc. Amer. Math. Soc. 143 (2015), no. 7,
2905-2912. (Reviewer: Stephan R. Garcia)
Previous Up Next
Citations
From References: 0
From Reviews: 0
MR3336615 (Review) 46E22 47B20 47B32
Hartz, Michael (3-WTRL-PM)
Nevanlinna-Pick spaces with hyponormal multiplication operators. (English
summary)
Proc. Amer. Math. Soc. 143 (2015), no. 7, 2905–2912.
The main result of this paper (Theorem 1.1) shows that the classical Hardy space H2
is essentially the only complete Nevanlinna-Pick space whose multiplication operators
are all hyponormal. To be more specific, if H is an irreducible complete Nevanlinna-
Pick space on a set X with kernel K such that all multiplication operators on H are
hyponormal, then either:
1. X is a singleton and H= C.
2. There is a set of uniqueness A ⊂ D for H2, a bijection j:X → A, and a nowhere
vanishing function δ:X → C such that
K(λ, µ) = δ(λ)δ(µ)k(j(λ), j(µ)).
Here k(z, w) denotes the Szego˝ kernel on H2.
In particular, the map f 7→ δ(f ◦ j) is a unitary operator. Continuity and analyticity are
also respected: if X is endowed with a topology such that K is separately continuous on
X ×X, then j is also continuous. If X ⊂ Cn and K is holomorphic in the first variable,
then j is holomorphic.
An interesting consequence of the main theorem is Corollary 1.3, which asserts that if
n≥ 3 and U ⊂ Rn is an open set, then there is no irreducible complete Nevanlinna-Pick
space on U which consists of continuous functions and whose multiplication operators
are all hyponormal. The proof appeals to Brouwer’s invariance of domain theorem.
Stephan R. Garcia
References
1. Jim Agler and John E. McCarthy, Complete Nevanlinna-Pick kernels, J. Funct.
Anal. 175 (2000), no. 1, 111–124, DOI 10.1006/jfan.2000.3599. MR1774853
(2001h:47019) MR1774853
2. Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces,
Graduate Studies in Mathematics, vol. 44, American Mathematical Society, Provi-
dence, RI, 2002. MR1882259
3. Ca˘lin-Grigore Ambrozie and Dan Timotin, On an intertwining lifting theorem
for certain reproducing kernel Hilbert spaces, Integral Equations Operator Theory
42 (2002), no. 4, 373–384, DOI 10.1007/BF01270919. MR1885440 (2002m:47012)
MR1885440
4. William Arveson, Subalgebras of C∗-algebras. III. Multivariable operator theory,
Acta Math. 181 (1998), no. 2, 159–228, DOI 10.1007/BF02392585. MR1668582
(2000e:47013) MR1668582
5. Joseph A. Ball, Tavan T. Trent, and Victor Vinnikov, Interpolation and commutant
lifting for multipliers on reproducing kernel Hilbert spaces, Operator theory and
analysis (Amsterdam, 1997), Oper. Theory Adv. Appl., vol. 122, Birkha¨user, Basel,
2001, pp. 89–138. MR1846055 (2002f:47028) MR1846055
6. L. E. J. Brouwer, Beweis der invarianz des n-dimensionalen gebiets (German), Math.
Ann. 71 (1911), no. 3, 305–313, DOI 10.1007/BF01456846. MR1511658 MR1511658
7. John B. Garnett, Bounded analytic functions, 1st ed., Graduate Texts in Mathe-
matics, vol. 236, Springer, New York, 2007. MR2261424 (2007e:30049) MR2261424
8. Allen L. Shields, Weighted shift operators and analytic function theory, Topics
in operator theory, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. Math.
Surveys, No. 13. MR0361899 (50 #14341) MR0361899
Note: This list reflects references listed in the original paper as accurately as
possible with no attempt to correct errors.
c© Copyright American Mathematical Society 2016


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Review: Nevanlinna-Pick spaces with hyponormal multiplication operators

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