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arXiv · 1905.00823

Universal Block Tridiagonalization in B(H) and Beyond

Sasmita Patnaik·Srdjan Petrovic·Gary Weiss
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Abstract

For H a separable infinite dimensional complex Hilbert space, we prove that every B(H) operator has a basis with respect to which its matrix representation has a universal block tridiagonal form with block sizes given by a simple exponential formula independent of the operator. From this, such a matrix representation can be further sparsified to slightly sparser forms; it can lead to a direct sum of even sparser forms reflecting in part some of its reducing subspace structure; and in the case of operators without invariant subspaces (if any exists), it gives a plethora of sparser block tridiagonal representations. An extension to unbounded operators occurs for a certain domain of definition condition. Moreover this process gives rise to many different choices of block sizes.

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Sasmita Patnaik, Srdjan Petrovic, Gary Weiss. 2019-05-02. Universal Block Tridiagonalization in B(H) and Beyond. https://arxiv.org/abs/1905.00823

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