## Rocky Mountain Journal of Mathematics

### Multivariable isometries related to certain convex domains

Ameer Athavale

#### Abstract

Several interesting results exist in the literature on subnormal operator tuples having their spectral properties tied to the geometry of strictly pseudoconvex domains or to that of bounded symmetric domains in $\mathbb{C} ^n$. We introduce a class $\Omega ^{(n)}$ of convex domains in $\mathbb{C} ^n$ which, for $n \geq 2$, is distinct from the class of strictly pseudoconvex domains and the class of bounded symmetric domains and which lends itself to the application of theories related to the abstract inner function problem and the $\overline \partial$-Neumann problem, allowing us to make a number of interesting observations about certain subnormal operator tuples associated with the members of the class $\Omega ^{(n)}$.

#### Article information

Source
Rocky Mountain J. Math., Volume 48, Number 1 (2018), 19-46.

Dates
First available in Project Euclid: 28 April 2018

https://projecteuclid.org/euclid.rmjm/1524880879

Digital Object Identifier
doi:10.1216/RMJ-2018-48-1-19

Mathematical Reviews number (MathSciNet)
MR3795731

Zentralblatt MATH identifier
06866698

Subjects
Primary: 47B20: Subnormal operators, hyponormal operators, etc.

#### Citation

Athavale, Ameer. Multivariable isometries related to certain convex domains. Rocky Mountain J. Math. 48 (2018), no. 1, 19--46. doi:10.1216/RMJ-2018-48-1-19. https://projecteuclid.org/euclid.rmjm/1524880879

#### References

• A.B. Aleksandrov, Inner functions on compact spaces, Funct. Anal. Appl. 18 (1984), 87–98.
• A. Athavale, Subnormal tuples quasi-similar to the Szegö tuple, Michigan Math. J. 35 (1988), 409–412.
• ––––, On the intertwining of joint isometries, J. Oper. Th. 23 (1990), 339–350.
• ––––, On the intertwining of $\partial {\mathcal D}$-isometries, Compl. Anal. Oper. Th. 2 (2008), 417–428.
• A. Athavale and S. Pedersen, Moment problems and subnormality, J. Math. Anal. Appl. 146 (1990), 434–441.
• A. Athavale and S. Podder, On the multiplication tuples related to certain reproducing kernel Hilbert spaces, Compl. Anal. Oper. Th. 10 (2016), 1329–1338.
• H.P. Boas and E.J. Straube, Global regularity of the $\overline \partial$-Neumann problem: A survey of the $L^2$-Sobolev theory, in Several complex variables, MSRI Publications 37 (1999), 79–111.
• E. Cartan, Sur les domaines bornés homogènes de l'espace des $n$ variables complexes, Abh. Math. Sem. Hamburg 11 (1935), 116–162.
• D.W. Catlin, Global regularity of the $\overline \partial$-Neumann problem, Proc. Sympos. Pure Math. 4 (1984), 39–49.
• J.B. Conway, The theory of subnormal operators, Math. Surv. Mono. 36, American Mathematical Society, Providence, RI, 1991.
• ––––, A course in operator theory, Grad. Stud. Math. 21, American Mathematical Society, Providence, RI, 2000.
• D. Crocker and I. Raeburn, Toeplitz operators on certain weakly pseudoconvex domains, J. Austral. Math. Soc. 31 (1981), 1–14.
• R.E. Curto, Spectral inclusion for doubly commuting subnormal $n$-tuples, Proc. Amer. Math. Soc. 83 (1981), 730–734.
• K.R. Davidson, On operators commuting with Toeplitz operators modulo the compact operators, J. Funct. Anal. 24 (1977), 356–368.
• M. Didas, Dual algebras generated by von Neumann $n$-tuples over strictly pseudoconvex sets, Dissert. Math. (Roz. Mat.) 425 (2004).
• ––––, A note on the Toeplitz projection associated with spherical isometries, preprint.
• M. Didas and J. Eschmeier, Subnormal tuples on strictly pseudoconvex and bounded symmetric domains, Acta Sci. Math. (Szeged) 71 (2005), 691–731.
• M. Didas, J. Eschmeier and K. Everard, On the essential commutant of analytic Toeplitz operators associated with spherical isometries, J. Funct. Anal. 261 (2011), 1361–1383.
• K. Diedrich and J.E. Fornaess, Pseudoconvex domains with real analytic boundary, Ann. Math. (1978), 371–384.
• J. Eschmeier, On the reflexivity of multivariable isometries, Proc. Amer. Math. Soc. 134 (2005), 1783–1789.
• J. Eschmeier and K. Everard, Toeplitz projections and essential commutants, J. Funct. Anal. 269 (2015), 1115–1135.
• G.B. Folland and J.J. Kohn, The Neumann problem for the Cauchy-Riemann complex, Ann. Math. Stud., Princeton University Press, Princeton, 1972.
• M. Grangé, Diviseurs de Leibenson et problème de Gleason pour $H^{\infty}(\Omega)$ dans le cas convexe, Bull. Soc. Math. France 114 (1986), 225–245.
• M. Hakim and N. Sibony, Frontiére de Shilov et spectre de $A(\overline D)$ pour les domaines faiblement psedoconvexes, C.R. Acad. Sci. Paris 281 (1975), 959–962.
• W.W. Hastings, Commuting subnormal operators simultaneously quasisimilar to unilateral shifts, Illinois J. Math. 22 (1978), 506–519.
• G.M. Henkin, The approximation of functions in pseudo-convex domains and a theorem of Z.L. Leibenzon, Bull. Acad. Polon. Sci. 19 (1971), 37–42.
• L. Hörmander, $L^2$ estimates and existence theorems for the $\overline \partial$ operator, Acta Math. 113 (1965), 89–152.
• T. Ito, On the commuting family of subnormal operators, J. Fac. Sci. Hokkaido Univ. 14 (1958), 1–15.
• M. Jarnicki and P. Pflug, First steps in several complex variables: Reinhardt domains, Europ. Math. Soc., 2008.
• N.P. Jewell and A.R. Lubin, Commuting weighted shifts and analytic function theory in several variables, J. Oper. Th. 1 (1979), 207–223.
• K. Kliś and M. Ptak, $k$-Hyperreflexive subspaces, Houston J. Math. 32 (2006), 299–313.
• J.J. Kohn, The range of the tangential Cauchy-Riemann operator, Duke Math. J. 53 (1986), 525–545.
• J.J. Kohn and H. Rossi, On the extension of holomorphic functions from the boundary of a complex manifold, Ann. Math. 81 (1965), 451–472.
• S. Krantz, Function theory of several complex variables, American Mathematical Society, Providence, RI, 2001.
• W. Mlak, Intertwining operators, Stud. Math. 43 (1972), 219–233.
• P. Pflug, Über polynomiale funktionen auf holomorphiegebieten, Math. Z. 138 (1974), 133–139.
• S.I. Pinchuk, Homogeneous domains with piecewise-smooth boundaries, Math. Z. 32 (1982) (in Russian); Math Notes 32 (1982), 849–852 (in English).
• B. Prunaru, Some exact sequences for Toeplitz algebras of spherical isometries, Proc. Amer. Math. Soc. 135 (2007), 3621–3630.
• M. Putinar, Spectral inclusion for subnormal $n$-tuples, Proc. Amer. Math. Soc. 90 (1984), 405–406.
• A.S. Raich and E.J. Straube, Compactness of the complex Green operator, Math. Res. Lett. 15 (2008), 761–778.
• R.M. Range, Holomorphic functions and integral representations in several complex variables, Springer-Verlag, New York, 1986.
• W. Rudin, New constructions of functions holomorphic in the unit ball of $\C^n$, CBMS Reg. Conf. Ser. Math., American Mathematical Society, Providence, RI, 1986.
• N. Salinas, The $\overline \partial$-formalism and the $\RC^*$-algebra of the Bergman $n$-tuple, J. Oper. Th. 22 (1989), 325–343.
• N. Salinas, A. Sheu and H. Upmeier, Toeplitz operators on pseudoconvex domains and foliation $\RC^*$-algebras, Ann. Math. 130 (1989), 531–565.
• A. Sheu, Isomprphism of the Toeplitz $\RC^*$-algebras for the Hardy and Bergman spaces of certain Reinhardt domains, Proc. Amer. Math. Soc. 116 (1992), 113–120.
• T. Sunada, Holomorphic equivalence problem for bounded Reinhardt domains, Math. Ann. 235 (1978), 111–128.
• J.L. Taylor, The analytic-functional calculus for several commuting operators, Acta Math. 125 (1970), 1–38.
• H. Upmeier, Toeplitz operators and index theory in several complex variables, Oper. Th. Adv. Appl. 81, Birkhäuser Verlag, Basel, 1996.