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Scalar-type kernels for block Toeplitz operators. (English) Zbl 1506.47046

Summary: It is shown that the kernel of a Toeplitz operator with \(2 \times 2\) symbol \(G\) can be described exactly in terms of any given function in a very wide class, its image under multiplication by \(G\), and their left inverses, if the latter exist. As a consequence, under many circumstances the kernel of a block Toeplitz operator may be described as the product of a space of scalar complex-valued functions by a fixed column vector of functions. Such kernels are said to be of scalar type, and in this paper they are studied and described explicitly in many concrete situations. Applications are given to the determination of kernels of truncated Toeplitz operators for several new classes of symbols.

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
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[1] Bart, H.; Tsekanovskiĭ, V.È., Matricial coupling and equivalence after extension, (Operator Theory and Complex Analysis. Operator Theory and Complex Analysis, Sapporo, 1991. Operator Theory and Complex Analysis. Operator Theory and Complex Analysis, Sapporo, 1991, Oper. Theory Adv. Appl., vol. 59 (1992), Birkhäuser: Birkhäuser Basel), 143-160 · Zbl 0807.47003
[2] Bessonov, R. V., Truncated Toeplitz operators of finite rank, Proc. Am. Math. Soc., 142, 4, 1301-1313 (2014) · Zbl 1314.47040
[3] Böttcher, A.; Karlovich, Yu. I.; Spitkovsky, I. M., Convolution Operators and Factorization of Almost Periodic Matrix Functions, Operator Theory: Advances and Applications, vol. 131 (2002), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 1011.47001
[4] Câmara, M. C.; Partington, J. R., Near invariance and kernels of Toeplitz operators, J. Anal. Math., 124, 235-260 (2014) · Zbl 1325.47061
[5] Câmara, M. C.; Partington, J. R., Spectral properties of truncated Toeplitz operators by equivalence after extension, J. Math. Anal. Appl., 433, 2, 762-784 (2016) · Zbl 1325.47062
[6] Câmara, M. C.; Partington, J. R., Asymmetric truncated Toeplitz operators and Toeplitz operators with matrix symbol, J. Oper. Theory, 77, 2, 455-479 (2017) · Zbl 1389.47080
[7] Câmara, M. C.; Partington, J. R., Multipliers and equivalences between Toeplitz kernels, J. Math. Anal. Appl., 465, 1, 557-570 (2018) · Zbl 1391.42010
[8] Câmara, M. C.; Partington, J. R., Toeplitz Kernels and Model Spaces. The Diversity and Beauty of Applied Operator Theory, Oper. Theory Adv. Appl., vol. 268, 139-153 (2018), Birkhäuser/Springer: Birkhäuser/Springer Cham
[9] Câmara, M. C.; Diogo, C.; Rodman, L., Fredholmness of Toeplitz operators and corona problems, J. Funct. Anal., 259, 5, 1273-1299 (2010) · Zbl 1205.47026
[10] Chalendar, I.; Chevrot, N.; Partington, J. R., Nearly invariant subspaces for backwards shifts on vector-valued Hardy spaces, J. Oper. Theory, 63, 2, 403-415 (2010) · Zbl 1224.46051
[11] Clancey, K.; Gohberg, I., Factorization of Matrix Functions and Singular Integral Operators (1981), Birkhäuser: Birkhäuser Basel-Boston-Stuttgart · Zbl 0474.47023
[12] Coburn, L. A., Weyl’s theorem for nonnormal operators, Mich. Math. J., 13, 285-288 (1966) · Zbl 0173.42904
[13] Crofoot, R. B., Multipliers between invariant subspaces of the backward shift, Pac. J. Math., 166, 2, 225-246 (1994) · Zbl 0819.47042
[14] Diogo, C., Problemas de Riemann-Hilbert com símbolos triangulares oscilatórios e factorizacã̧o generalizada (2004), Instituto Superior Técnico, Master thesis
[15] Fricain, E.; Hartmann, A.; Ross, W. T., Multipliers between model spaces, Stud. Math., 240, 2, 177-191 (2018) · Zbl 1394.30043
[16] Fricain, E.; Hartmann, A.; Ross, W. T., Range spaces of co-analytic Toeplitz operators, Can. J. Math., 70, 6, 1261-1283 (2018) · Zbl 1418.30051
[17] Garnett, J. B., Bounded Analytic Functions, Graduate Texts in Mathematics, vol. 236 (2007), Springer: Springer New York
[18] Gu, C.; Kang, D.-O., Rank of truncated Toeplitz operators, Complex Anal. Oper. Theory, 11, 4, 825-842 (2017), (English summary) · Zbl 1454.47037
[19] Hartmann, A.; Mitkovski, M., Kernels of Toeplitz Operators. Recent Progress on Operator Theory and Approximation in Spaces of Analytic Functions, Contemp. Math., vol. 679, 147-177 (2016), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1375.30084
[20] Hitt, D., Invariant subspaces of \(\mathcal{H}^2\) of an annulus, Pac. J. Math., 134, 1, 101-120 (1988) · Zbl 0662.30035
[21] Koosis, P., Introduction to \(H_p\) Spaces (1998), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1024.30001
[22] Mikhlin, S. G.; Prössdorf, S., Singular Integral Operators. Translated from the German by Albrecht Böttcher and Reinhard Lehmann (1986), Springer-Verlag: Springer-Verlag Berlin · Zbl 0612.47024
[23] Nikolski, N. K., Operators, Functions, and Systems: An Easy Reading. Vol. 1. Hardy, Hankel, and Toeplitz, Mathematical Surveys and Monographs, vol. 92 (2002), American Mathematical Society: American Mathematical Society Providence, RI, Translated from the French by Andreas Hartmann · Zbl 1007.47001
[24] O’Loughlin, R., Transfer Report (2018), University of Leeds
[25] Sarason, D., Nearly invariant subspaces of the backward shift, (Contributions to Operator Theory and Its Applications. Contributions to Operator Theory and Its Applications, Mesa, AZ, 1987. Contributions to Operator Theory and Its Applications. Contributions to Operator Theory and Its Applications, Mesa, AZ, 1987, Oper. Theory Adv. Appl., vol. 35 (1988), Birkhäuser: Birkhäuser Basel), 481-493 · Zbl 0687.47003
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