6th Summer Workshop on Operator Theory

9th - 13th July 2018 Kraków

Patryk Pagacz

Invariant subspaces of H2(T2) and L2(T2) preserving compatibility
We consider the operators Tw, Tz of multiplication by independent variables “w”,”z” on the space of square summable functions over the torus and its Hardy subspace.
We say that an invariant subspace M preserves compatibility if a pair (Tw|M,Tz|M) is compatible, i.e.

PTwnMPTzmM=PTzmMPTwnM, for any n,m ∈ N.

We describe an invariant subspaces of Hardy space H2(T2) which preserves compatibility as φMJ, where φ is an inner function and MJ is a subspace (i,j)∈ J wizj, with some cone J⊂ N2.

Moreover, we give a full description of invariant subspaces of L2(T2) preserving compatibility.

The talk is based on joint work with Zbigniew Burdak, Marek Kosiek and Marek Słociński.

  1. Z. Burdak, M. Kosiek, P. Pagacz and M. Słociński: Invariant subspaces of H2(T2) andL2(T2) preserving compatibility, J. Math. Anal. Appl., 455 (2017), 1706–1719.
  2. Z. Burdak, M. Kosiek and M. Słociński: Compatible pairs of commuting isometries, Linear Algebra Appl., 479 (2015), 216–259.
  3. Z. Burdak, M. Kosiek, P. Pagacz and M. Słociński: On the commuting isometries, Linear Algebra Appl., 516 (2017), 167-185.
  4. V. Mandrekar: The validity of beurling theorems in polidiscs, Proc. Amer. Math. Soc., 103 (1988), 145–148.
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