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Zero products of Toeplitz operators. (English) Zbl 1170.47013

Let \(H^p\), \(1<p<\infty\), be the Hardy space of the unit circle \({\mathbb T}\) and \(H_m^p\) be the space of all vector-valued functions \(f=(f^1,\dots,f^m)\), where each \(f^j\in H^p\). Further, let \(L^\infty({\mathbb T },M_{m\times m}({\mathbb C}))\) denote the set of all \(m\times m\) matrices with essentially bounded entries. By \(T_u\) denote the Toeplitz operator on \(H^p_m\) with symbol \(u\in L^\infty({\mathbb T},M_{m\times m}({\mathbb C}))\). The main result of the paper says that if \(u_1,\dots,u_n\in L^\infty({\mathbb T},M_{m\times m}({\mathbb C}))\) and \(T_{u_1}\cdots T_{u_n}=0\), then \(\det u_k=0\) almost everywhere on \({\mathbb T}\) for some \(k\in\{1,\dots,n\}\).

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
30H05 Spaces of bounded analytic functions of one complex variable
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[1] P. Ahern and ž. čUčKović, A theorem of Brown-Halmos type for Bergman space Toeplitz operators , J. Funct. Anal. 187 (2001), 200–210. · Zbl 0996.47037 · doi:10.1006/jfan.2001.3811
[2] S. Axler, personal communication, October 2000.
[3] A. Brown and P. R. Halmos, Algebraic properties of Toeplitz operators , J. Reine Angew. Math. 213 (1963/1964), 89–102. · Zbl 0116.32501
[4] J. Cima, A. Matheson, and W. T. Ross, The Cauchy Transform , Math. Surveys Monogr. 125 , Amer. Math. Soc., Providence, 2006. · Zbl 1096.30046
[5] R. G. Douglas, Banach Algebra Techniques in the Theory of Toeplitz Operators (Athens, Ga., 1972) , CBMS Regional Conf. Ser. in Math. 15 , Amer. Math. Soc., Providence, 1973. · Zbl 0252.47025
[6] C. Gu, Products of several Toeplitz operators , J. Funct. Anal. 171 (2000), 483–527. · Zbl 0967.47021 · doi:10.1006/jfan.1999.3547
[7] K. Y. Guo, A problem on products of Toeplitz operators , Proc. Amer. Math. Soc. 124 (1996), 869–871. JSTOR: · Zbl 0841.47015 · doi:10.1090/S0002-9939-96-03224-8
[8] D. Hitt, Invariant subspaces of \(\mathscrH^2\) of an annulus , Pacific J. Math. 134 (1988), 101–120. · Zbl 0662.30035 · doi:10.2140/pjm.1988.134.101
[9] P. Koosis, Introduction to \(H_p\) Spaces , 2nd ed., Cambridge Tracts in Math. 115 , Cambridge Univ. Press, Cambridge, 1998. · Zbl 1024.30001
[10] D. Sarason, “Nearly invariant subspaces of the backward shift” in Contributions to Operator Theory and Its Applications, (Mesa, Ariz., 1987) , Oper. Theory Adv. Appl. 35 , Birkhäuser, Basel, 1988, 481–493. · Zbl 0687.47003
[11] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals , Princeton Math. Ser. 43 , Princeton Univ. Press, Princeton, 1993. · Zbl 0821.42001
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