On the interplay between operators, bases, and matrices

In honor of N.K. Nikolski on the occasion of his eightieth anniversary
https://doi.org/10.1016/j.jfa.2021.109158Get rights and content

Abstract

Given a bounded linear operator T on a separable Hilbert space, we develop an approach allowing one to construct a matrix representation for T having certain specified algebraic or asymptotic structure. We obtain matrix representations for T with preassigned bands of the main diagonals, with an upper bound for all of the matrix elements, and with entrywise rational-like lower and upper bounds for these elements. In particular, we substantially generalize and complement our results on diagonals of operators from [47] and other related results. Moreover, we obtain a vast generalization of a theorem by Stout (1981) [56], and (partially) answer his open question. Several of our results have no analogues in the literature.

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 matrixAT:=(Tun,uj)j,n=1 with respect to an orthonormal basis (un)n=1H and try to relate the properties of T to the properties of AT. 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 W(T):={Tx,x:x=1} and its essential numerical range We(T). The main property of W(T) is that it is always a convex subset of C, and moreover the spectrum σ(T) of T is contained in the closure W(T) of W(T). However, W(T) can be neither closed nor open.

Being only an approximate analogue of W(T), 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 TB(H) 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 (ym)m=1 be a sequence of unit vectors in H such that m=1ym=H.

For r=0,1,,K let Br={nN:n=rmod(K+1)}, and note that there exists r0{0,,K} such that nBr0dist{λn,We(T)}=.

Represent Br0 as Br0=m=1Am,

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 TB(H) 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 (an)n=1(0,) be such (an)n=11(N). Then there exists (an)n=1(0,) such that (an)n=11(N), 0<anmax{1,an} for all nN, and limnanan=0.

Proof

Set n0=0. We construct numbers nk,kN

Matrices with large entries: proofs

In this section we proceed with the proof of Theorem 2.8 producing a matrix with large entries for TB(H) with We(T) 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 TB(H) is compact if and only if We(T)={0}. So T is of the form T=λI+K for some λC and a compact operator KB(H) if and only if We(T) is a singleton, i.e., the diameterdiam(We(T))=max{|λμ|:λ,μWe(T)}=0.

Lemma 5.1

Let TB(H) 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 T=(T1,,Tm)B(H)m, mN. Moreover, their versions for tuples of selfadjoint operators can be formulated under more general assumptions by replacing the interior of the essential numerical range We(T) 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.

References (60)

  • W. Arveson

    Diagonals of normal operators with finite spectrum

    Proc. Natl. Acad. Sci. USA

    (2007)
  • W. Arveson et al.

    Diagonals of Self-Adjoint Operators, Operator Theory, Operator Algebras, and Applications

    (2006)
  • K. Ball

    Convex geometry and functional analysis

  • K.M. Ball

    The complex plank problem

    Bull. Lond. Math. Soc.

    (2001)
  • G. Bennett

    Schur multipliers

    Duke Math. J.

    (1977)
  • F.F. Bonsall et al.

    Numerical Ranges, II

    (1973)
  • J.-C. Bourin

    Compressions and pinchings

    J. Oper. Theory

    (2003)
  • M. Bownik et al.

    Characterization of sequences of frame norms

    J. Reine Angew. Math.

    (2011)
  • M. Bownik et al.

    The Schur-Horn theorem for operators with finite spectrum

    Trans. Am. Math. Soc.

    (2015)
  • A. Brown et al.

    Structure of commutators of operators

    Ann. Math.

    (1965)
  • N.P. Brown et al.

    C-Algebras and Finite-Dimensional Approximations

    (2008)
  • A.T. Dash

    Joint essential spectra

    Pac. J. Math.

    (1976)
  • M. David

    On a certain type of commutator

    J. Math. Mech.

    (1970)
  • M. David

    On a certain type of commutators of operators

    Isr. J. Math.

    (1971)
  • K.R. Davidson et al.

    Norms of Schur multipliers

    Ill. J. Math.

    (2007)
  • R.G. Douglas et al.

    A note on quasitriangular operators

    Duke Math. J.

    (1970)
  • P. Fan

    On the diagonal of an operator

    Trans. Am. Math. Soc.

    (1984)
  • P. Fan et al.

    On zero-diagonal operators and traces

    Proc. Am. Math. Soc.

    (1987)
  • P.A. Fillmore et al.

    On the essential numerical range, the essential spectrum, and a problem of Halmos

    Acta Sci. Math. (Szeged)

    (1972)
  • C.K. Fong et al.

    Diagonal operators: dilation, sum and product

    Acta Sci. Math. (Szeged)

    (1993)
  • 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.

    View full text