Mixed Invariant Subspaces over the Bidisk |
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Authors: | Kei Ji Izuchi Kou Hei Izuchi Masatoshi Naito |
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Institution: | 1. Department of Mathematic, Niigata University, Niigata, 950-2181, Japan 2. Department of Mathematic, Korea University, Seoul, 136-701, Korea 3. Graduate School of Science and Technology, Niigata University, Niigata, 950-2181, Japan
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Abstract: | It is known that the structure of invariant subspaces I of the Hardy space H 2 over the bidisk is extremely complicated. One reason is that it is difficult to describe infinite dimensional wandering spaces ${I\ominus zI}$ completely. In this paper, we study the structure of nontrivial closed subspaces N of H 2 with ${T_zN\subset N}$ and ${T^*_wN\subset N}$ , which are called mixed invariant subspaces under T z and ${T^*_w}$ . We know that the dimension of ${N\ominus zN}$ ranges from 1 to ??. If ${T^*_w(N\ominus zN)\subset N\ominus zN}$ , we may describe N completely. If ${T^*_w(N\ominus zN)\not\subset N\ominus zN}$ , it seems difficult to describe N generally. So we study N under the condition ${dim\,(N\ominus zN)=1}$ . Write ${M=H^2\ominus N}$ . We describe ${M\ominus wM}$ precisely. We give a characterization of N for which there is a nonzero function ${\varphi}$ in ${M\ominus wM}$ satisfying ${z^k\varphi\in M\ominus wM}$ for every k ?? 0. We also see that the space ${M\ominus wM}$ has a deep connection with the de Branges?CRovnyak spaces studied by Sarason. |
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