On the interplay between operators, bases, and matrices☆
Section snippets
Introduction: a glimpse at matrix representations
Following conventional approach to describing operators on finite-dimensional spaces as matrices, one may represent a bounded linear operator T on infinite-dimensional separable Hilbert space H as the matrix with respect to an orthonormal basis and try to relate the properties of T to the properties of . This very natural idea looks naive to some extent, and the study of operators on infinite-dimensional spaces through their matrix representations goes back to
Numerical ranges
In our studies of matrix representations for a bounded operator T on a separable (complex) infinite-dimensional Hilbert space H we will rely on elementary properties of its numerical range and its essential numerical range . The main property of is that it is always a convex subset of , and moreover the spectrum of T is contained in the closure of . However, can be neither closed nor open.
Being only an approximate analogue of , the
Matrices with several given diagonals: proofs
We start with a proof of Theorem 2.2. The result provides a generalization of [47, Corollary 4.3] in case of a single operator by describing a large set of the main diagonals for T and arranging arbitrary bands of zeros around them. (Concerning the setting of tuples see Section 6.)
Proof of Theorem 2.2 Let be a sequence of unit vectors in H such that . For let , and note that there exists such that . Represent as ,
Matrices with small entries: proofs
In this section, we extend Stout's bound from Theorem 1.3, (iii) by providing a similar bound for all matrix elements of rather than merely the main diagonal of T for an appropriate basis in H. Clearly, this will also significantly improve Theorem 1.3, (ii). To this aim, we need a simple lemma.
Lemma 4.1 Let be such . Then there exists such that , for all , and .
Proof Set . We construct numbers
Matrices with large entries: proofs
In this section we proceed with the proof of Theorem 2.8 producing a matrix with large entries for with containing more than two points.
We will need the next lemma similar in spirit to considerations in [11, Section 2], see also [51]. Recall that is compact if and only if . So T is of the form for some and a compact operator if and only if is a singleton, i.e., the diameter
Lemma 5.1 Let be an operator which is
The setting of operator tuples
Note that Theorem 2.2, Theorem 2.4, Theorem 2.5 have their counterparts for tuples of bounded linear operators , . Moreover, their versions for tuples of selfadjoint operators can be formulated under more general assumptions by replacing the interior of the essential numerical range with its appropriate relative interior. We have decided to present their single operator versions to simplify the presentation and to illustrate the method rather than its fine
Acknowledgement
We would like to thank the referee for carefully reading the manuscript and helpful suggestions improving the presentation.
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The first author has been supported by grant No. 20-31529X of GACR and RVO: 67985840. The second author was partially supported by NCN grant UMO-2017/27/B/ST1/00078.