**Invariant subspaces of H ^{2}(T^{2}) and L^{2}(T^{2}) preserving compatibility**

We consider the operators T

_{w}, T

_{z}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 (T

_{w}|

_{M},T

_{z}|

_{M}) is compatible, i.e.

P_{TwnM}P_{TzmM}=P_{TzmM}P_{TwnM}, for any n,m ∈ N.

We describe an invariant subspaces of Hardy space H^{2}(T^{2}) which preserves compatibility as φM_{J}, where φ is an inner function and M_{J} is a subspace ⋁_{(i,j)∈ J} w^{i}z^{j}, with some cone J⊂ N^{2}.

Moreover, we give a full description of invariant subspaces of L^{2}(T^{2}) preserving compatibility.

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

- Z. Burdak, M. Kosiek, P. Pagacz and M. Słociński: Invariant subspaces of H
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*Linear Algebra Appl*., 479 (2015), 216–259. - Z. Burdak, M. Kosiek, P. Pagacz and M. Słociński: On the commuting isometries,
*Linear Algebra Appl*., 516 (2017), 167-185. - V. Mandrekar: The validity of beurling theorems in polidiscs,
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