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Curvature inequalities and extremal operators. (English) Zbl 1429.30017

For a bounded domain \(\Omega\subset\mathbb{C}^m\), let \(\Omega^*=\{\overline{z}:z\in\Omega\}\). An \(m\)-tuple \(\mathbf{T}=(T_1,T_2,\ldots,T_m)\) of commuting bounded operators on a complex separable Hilbert space \(\mathcal{H}\) is said to be in \(B_n(\Omega^*)\) if:
(1)
\(\dim\big(\bigcap_{k=1}^m\ker(T_k-w_kI)\big)=n\) for each \(w=(w_1,w_2,\ldots,w_m)\in\Omega^*\);
(2)
the operator \(\mathcal{D}_{\mathbf{T}-w\mathbf{I}} =\bigoplus_{k=1}^m(T_k-w_kI)\), \(w\in\Omega^*\), has closed range in the Hilbert space \(\bigoplus_{k=1}^m\mathcal{H}\);
(3)
\(\bigvee_{w\in\Omega^*}\big(\bigcap_{k=1}^m\ker(T_k-w_kI)\big)=\mathcal{H}\).

For any commuting tuple of operators \(\mathbf{T}\) in \(B_n(\Omega^*)\), there exists a rank-\(n\) holomorphic Hermitian vector bundle \(E_T\) over \(\Omega^*\), \[ E_T:=\Big\{(w,v)\in\Omega^*\times\mathcal{H}:\ v\in\bigcap_{k=1}^m\ker(T_k-w_kI)\Big\}. \] For \(w\in\Omega\), properties of the curvature \(\mathcal{K}(w)\) of the vector bundle \(E_T\) are investigated. For contractive commuting tuples of operators \(\mathbf{T}\in B_n(\Omega^*)\), a curvature inequality is established. The properties of the extremal operators transforming the curvature inequality into the equality are studied. For the open unit disc \(\mathbb{D}\), the unitary equivalence of a contraction \(T\in B_1(\mathbb{D})\) and the backward shift operator \(U_+^*\) is proved under two additional conditions.

MSC:

30C40 Kernel functions in one complex variable and applications
47A13 Several-variable operator theory (spectral, Fredholm, etc.)
47A25 Spectral sets of linear operators
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References:

[1] M. B. Abrahamse and R. G. Douglas, A class of subnormal operators related to multiply-connected domains, Adv. Math. 19 (1976), no. 1, 106-148. · Zbl 0321.47019 · doi:10.1016/0001-8708(76)90023-2
[2] J. Agler, Hypercontractions and subnormality, J. Operator Theory 13 (1985), no. 2, 203-217. · Zbl 0593.47022
[3] S. R. Bell, “The Cauchy transform, potential theory, and conformal mapping” in Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.
[4] M. J. Cowen and R. G. Douglas, Complex geometry and operator theory, Acta Math. 141 (1978), no. 3-4, 187-261. · Zbl 0427.47016 · doi:10.1007/BF02545748
[5] M. J. Cowen and R. G. Douglas, “Operators possessing an open set of eigenvalues,” in Functions, Series, Operators, Vol. I, II (Budapest, 1980), Colloq. Math. Soc. János Bolyai, Vol. 35, North Holland, 1983, 323-341. · Zbl 0584.47002
[6] R. E. Curto and N. Salinas, Generalized Bergman kernels and the Cowen-Douglas theory, Amer. J. Math. 106 (1984), no. 2, 447-488. · Zbl 0583.47037 · doi:10.2307/2374310
[7] R. G. Douglas, D. K. Keshari, and A. Xu, Generalized bundle shift with application to multiplication operator on the Bergman space, J. Operator Theory 75 (2016), no. 1, 3-19. · Zbl 1389.47068 · doi:10.7900/jot.2014sep05.2051
[8] S. D. Fisher, “Function theory on planar domains” in Pure and Applied Mathematics, Wiley & Sons, New York, 1983. · Zbl 0511.30022
[9] G. Misra, Curvature and the backward shift operators, Proc. Amer. Math. Soc. 91 (1984), no. 1, 105-107. · Zbl 0548.47015
[10] G. Misra, Curvature inequalities and extremal properties of bundle shifts, J. Operator Theory 11 (1984), no. 2, 305-317. · Zbl 0544.47015
[11] G. Misra and A. Pal, Contractivity, complete contractivity and curvature inequalities, J. Anal. Math. 136, (2018), 31-54. · Zbl 1475.47017 · doi:10.1007/s11854-018-0054-7
[12] G. Misra and V. Pati, Contractive and completely contractive modules, matricial tangent vectors and distance decreasing metrics, J. Operator Theory 30 (1993), 353-380. · Zbl 0833.46043
[13] G. Misra and N. S. N. Sastry, Bounded modules, extremal problems, and a curvature inequality, J. Funct. Anal. 88 (1990), no. 1, 118-134. · Zbl 0727.46029 · doi:10.1016/0022-1236(90)90121-Z
[14] Z. Nehari, Conformal Mapping, Dover, New York, 1975. · Zbl 0048.31503
[15] M. R. Reza, Curvature inequalities for operators in the Cowen-Douglas class of a planar domain, Indiana Univ. Math. J. 67 (2018), no. 3, 1255-1279. · Zbl 06971418 · doi:10.1512/iumj.2018.67.7320
[16] S. Shimorin, Wold-type decompositions and wandering subspaces for operators close to isometries, J. Reine Angew. Math. 531 (2001), 147-189. · Zbl 0974.47014
[17] N. Suita, On a metric induced by analytic capacity. II, Kōdai Math. Sem. Rep. 27 (1976), nos. 1-2, 159-162. · Zbl 0335.30003 · doi:10.2996/kmj/1138847170
[18] M. Uchiyama, Curvatures and similarity of operators with holomorphic eigenvectors, Transactions of the American Mathematical Society 319 (1990), no. 1, 405-415. · Zbl 0733.47015 · doi:10.1090/S0002-9947-1990-0968421-4
[19] M. Voichick, Ideals and invariant subspaces of analytic functions, Trans. Amer. Math. Soc. 111 (1964), 493-512. · Zbl 0147.11502 · doi:10.1090/S0002-9947-1964-0160920-5
[20] K. Wang and G. Zhang, Curvature inequalities for operators of the Cowen-Douglas class, Israel J. Math. 222 (2017), no. 1, 279-296. · Zbl 1497.47034 · doi:10.1007/s11856-017-1590-z
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