Block Toeplitz Operators on Vector-Valued Doubling Fock Spaces
DOI:
https://doi.org/10.22401/ANJS.29.2.14Keywords:
Block Toeplitz operators , Vector-valued doubling Fock spaces , Matrix-valued symbol , Fredholm propertiesAbstract
This paper intends to examine the identity of the block Toeplitz operator 〖BT〗_Υ^m, with a matrix-valued symbol Υ, on vector-valued doubling Fock spaces F_(ϑ,m)^p. For an integer m, the spaces F_(ϑ,m)^p are characterized as the spaces of all vector functions F whose component functions belong to the doubling Fock space F_ϑ^p, and the operator 〖BT〗_Υ^m is defined upon the F_(ϑ,m)^p. However, fundamental properties like boundedness, compactness of 〖BT〗_Υ^m are explored. Additionally, for a specific type of matrix-valued symbol Υ, the necessary and sufficient condition on the symbol Υ that makes 〖BT〗_Υ^m Fredholm on F_(ϑ,m)^p are established.
References
[1] Bauer, W.; Coburn, L. A.; Isralowitz, J.; "Heat flow, BMO, and the compactness of Toeplitz operators". J. Funct. Anal., 259(1): 57-78, 2010.
[2] Basor E. L.; Ehrhardt T.; "Asymptotics of block Toeplitz determinants and the classical Dimer model". Commun. Math. Phys., 274: 427-455, 2006.
[3] Böttcher, A.; Silbermann, B.; "Analysis of Toeplitz operators". 2nd. ed.; Springer Monographs in Mathematics: Verlag, Berlin, 2010.
[4] Gu, C.; Hendricks, J.; Rutherford, D.; "Hyponormality of Block Toeplitz Operators". Pac. J. Math., 223(1): 95-111, 2006.
[5] Curto, R. E.; Hwang, I. S.; Lee, W. Y.; "Hyponormality and Subnormality of Block Toeplitz Operators". Adv. Math., 230(4-6): 2094-2151, 2012.
[6] Hwang, I. S.; Lee, W. Y.; "Block Toeplitz operators with rational symbols". J. Phys. A: Math. Theor., 41(18): 185-207, 2008.
[7] Lee, J.; "Hyponormality of block Toeplitz operators on the weighted Bergman spaces". Acta Math. Sci., 37(6): 1695-1704, 2017.
[8] Al-Qabani, A.; Virtanen, J.A.; "Fredholm theory of Toeplitz operators on standard weighted Fock spaces". Ann. Acad. Sci. Fenn. Math., 43:769-783, 2018.
[9] Oliver, R. & Pascuas, D.; "Toeplitz operators on doubling Fock spaces". J. Math. Anal. Appl., 435(2): 1426-1457, 2016.
[10] Marzo, J.; Ortega-Cerd, J.; "Pointwise estimates for the Bergman kernel of the weighted Fock space". J. Geom. Anal., 19(4): 890-910, 2009.
[11] Marphy, G. J.; "C*-algebras and operator theory". 1st. ed.; Academic Press, London, UK, 1990.
[12] Al-Qabani, A.; Hilberdink, T.; Virtanen, J. A.; "Fredholm theory of Toeplitz operators on doubling Fock Hilbert spaces". Math. Scand., 126(3): 593–602, 2020.
[13] Zhu, K.; “Analysis on Fock Spaces”. 1st. ed.; Graduate Texts in Mathematics, Springer: New York, NY, USA, 2012.
[14] Krupnik, N.Y.; “Banach Algebras with Symbol and Singular Integral Operators”. 1st. ed.; Operator Theory: Advances and Applications, Birkhäuser: Basel, Switzerland, 1987.
[15] Hu, Z.; Virtanen, J. A.; "Fredholm Toeplitz Operators on Doubling Fock Spaces". J. Geom. Anal., 32:160, 2022.
[16] Muhammad A.R.; "On the Solvability of Operator Equation AT+T*B=C". Al-Nahrain J. Sci., 15(2): 156-160, 2012.
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Copyright (c) 2026 Aamena Al-Qabani, Shayma Abdulsattar
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