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When products of projections diverge. (English) Zbl 1462.46021

For a fixed natural number \(K\), suppose that \(L_1,L_2,\dots,L_K\) is a family of \(K\) closed subspaces of a Hilbert space \(H\) such that \(\bigcap_{i=1}^K L_i=\{0\}\). Let \(z_0\in H\) and \(k_1,k_2,\ldots \in [K]=\{1,2,\dots,K\}\) be an arbitrary sequence. Let \(\{z_n\}_{n=1}^{\infty}\) be the following sequence \(z_{n}=P(L_{k_n})z_{n-1}\), where \(P(L_{k_n})\) denotes the orthogonal projection of \(H\) onto the subspace \(L_{k_n}\). If the sequence \(\{k_n\}\) is periodic, the product of the projections is called cyclic. The convergence is, however, either exponentially fast or arbitrarily slow, depending on whether \(L_1^{\perp}+\dots+L_K^{\perp}\) is closed or not.
In this paper, the author proves how slow convergence of cyclic iterates of projections corresponds to existence of non-converging iterates. More precisely, she considers subspaces \(L_1,\dots,L_K\) for which there is a non-convergent sequence of products of projections. She uses Johnson graphs to describe the family of sets of indices \(A\subseteq [K]\) for which \(\sum_{i\in A} L_{i}^{\perp}\) is not closed or \(\bigcap_{i\in A}L_{i}\) is infinite dimensional, and answers a question of F. Deutsch and H. Hundal [J. Approx. Theory 162, No. 9, 1717–1738 (2010; Zbl 1208.41013)]. Let \(K\geq 3\) and let \(L_1^{\perp}+ \dots +L_K^{\perp}\) be not closed. The author constructs \(\tilde L_k\subseteq L_k\), \(k\in [K]\), and a product of projections on the spaces \(\tilde L_k\) which does not converge in the norm. In addition, for \(K=3\) and \(K=4\), she proves that the existence of such \(\tilde L_k\)’s is equivalent to \(L_1^{\perp}+\dots +L_K^{\perp}\) not being closed. Some examples are presented to illustrate the investigation.

MSC:

46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
05C38 Paths and cycles

Citations:

Zbl 1208.41013
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References:

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