Analytic Toeplitz operators with automorphic symbol
Analytic Toeplitz operators with automorphic symbol
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DOI：
10.1090/s0002-9939-1975-0405156-8
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发表时间：
1975
期刊：
影响因子：
--
通讯作者：
M. Abrahamse
中科院分区：
文献类型：
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作者：
M. Abrahamse
Let R denote the annulus jz: 'A < lzl < 1} and let iT be a holomorphic universal covering map from the unit disk onto R. It is showrn that if iT is a function of an inner function &, that is, if 7T(z) = iT(w(z)), then w is a linear fractional transformation. However, the analytic Toeplitz operator T,r has nontrivial reducing subspaces. These facts answer in the negative a question raised by Nordgren [10]. Let k be the function q5(z) = T(Z)34 and let = xF be the inner-outer factorization of t. An operator C is produced which commutes with T but does not commute with TX nor with TF. This answers in the negative a question raised by Deddens and Wong [7]. The functions iT and k are both automorphic under the group of covering transformations for iT and hence may be viewed as functions on the annulus R. This point of view is critical in these examples. Let D denote the open unit disk, let H2 denote the Hardy space of functions / analytic on D with fI/(reZO )12 dO bounded independent of r, let Hm be the space of bounded analytic functions on D, and for 0 in Hm let To denote the operator on H2 defined by T+(/) =b/. The operator T? is said to be an analytic Toeplitz operator. A function c in Hm is said to be inner if limr_i r w(reZo)l = 1 for almost every 0 on the unit circle. If X is inner and nonconstant, then the analytic Toeplitz operator T. is a unilateral shift. Moreover, this unilateral shift T. has multiplicity one if and only if X is a linear fractional transformation. If co is not a linear fractional transformation, then T. is a shift of multiplicity greater than one and has therefore many nontrivial reducing subspaces. Moreover, if S in Hm is a function of w, that is, if 0(z)Vi(o(z)) for some Vi in Hm, then any reducing subspace for T. also reduces T?, [10, Theorem 2]. Thus, if S is a function of an inner function which is not a linear fractional transformation, then T has nontrivial reducingsubspaces. In [10], E. Nordgren inquires about the converse. Question 1. If the analytic Toeplitz operator T?, has nontrivial reducing subspaces, must there be an inner function w which is not a linear fractional transformation and a function Vi in Hm such that =(z)V(W(z)) for all z in D? Received by the editors June 7, 1974 and, in revised form, September 5, 1974. AMS (MOS) subject classifications (1970). Primary 47B35, 47B20, 30A58.
Let R denote the annulus jz: 'A < lzl < 1} and let iT be a holomorphic universal covering map from the unit disk onto R. It is showrn that if iT is a function of an inner function &, that is, if 7T(z) = iT(w(z)), then w is a linear fractional transformation. However, the analytic Toeplitz operator T,r has nontrivial reducing subspaces. These facts answer in the negative a question raised by Nordgren [10]. Let k be the function q5(z) = T(Z)34 and let = xF be the inner-outer factorization of t. An operator C is produced which commutes with T but does not commute with TX nor with TF. This answers in the negative a question raised by Deddens and Wong [7]. The functions iT and k are both automorphic under the group of covering transformations for iT and hence may be viewed as functions on the annulus R. This point of view is critical in these examples. Let D denote the open unit disk, let H2 denote the Hardy space of functions / analytic on D with fI/(reZO )12 dO bounded independent of r, let Hm be the space of bounded analytic functions on D, and for 0 in Hm let To denote the operator on H2 defined by T+(/) =b/. The operator T? is said to be an analytic Toeplitz operator. A function c in Hm is said to be inner if limr_i r w(reZo)l = 1 for almost every 0 on the unit circle. If X is inner and nonconstant, then the analytic Toeplitz operator T. is a unilateral shift. Moreover, this unilateral shift T. has multiplicity one if and only if X is a linear fractional transformation. If co is not a linear fractional transformation, then T. is a shift of multiplicity greater than one and has therefore many nontrivial reducing subspaces. Moreover, if S in Hm is a function of w, that is, if 0(z)Vi(o(z)) for some Vi in Hm, then any reducing subspace for T. also reduces T?, [10, Theorem 2]. Thus, if S is a function of an inner function which is not a linear fractional transformation, then T has nontrivial reducingsubspaces. In [10], E. Nordgren inquires about the converse. Question 1. If the analytic Toeplitz operator T?, has nontrivial reducing subspaces, must there be an inner function w which is not a linear fractional transformation and a function Vi in Hm such that =(z)V(W(z)) for all z in D? Received by the editors June 7, 1974 and, in revised form, September 5, 1974. AMS (MOS) subject classifications (1970). Primary 47B35, 47B20, 30A58.