Diagonality and Idempotents with applications to problems in operator theory and frame theory

Abstract

Abstract. We prove that a nonzero idempotent is zero-diagonal if and only if it is not a Hilbert-Schmidt perturbation of a projection, along with other useful equivalences. Zero-diagonal operators are those whose diagonal entries are identically zero in some basis.We also prove that any bounded sequence appears as the diagonal of some idempotent operator, thereby providing a characterization of inner products of dual frame pairs in infinite dimensions. Furthermore, we show that any absolutely summable sequence whose sum is a positive integer appears as the diagonal of a finite rank idempotent.

Diagonals of operators and Blaschke’s enigma

“…As remarked in [25], the numerical range results allow one to deduce easily Fong's theorem from [23] saying that every bounded sequence (d n ) ∞ n=1 admits a nilpotent operator (of index 2) whose diagonal is (d n ) ∞ n=1 . Moreover, in the same way, one can derive a similar result from [38] replacing nilpotents by idempotents. See Remark 4.4 for more on that.…”

“…To put our consideration into a proper framework, let us first review major achievements made so far in the study of diagonals of operators on infinite-dimensional spaces. First, note that (as mentioned in [38]) the study of diagonals can be understood in at least two senses:…”

“…in [21] and [20]. By different methods, it has been shown recently in [38] that an infinite-rank idempotent admits a zero-diagonal if and only if it is not a Hilbert-Schmidt perturbation of a (selfadjoint) projection.…”

Joint numerical ranges and compressions of powers of operators

“…As a consequence, we obtain a partial generalization of the ‘pinching’ theorem by Bourin [, Theorem 2.1] in a much more demanding setting of operator tuples. For recent applications of [, Theorem 2.1] see …, and consult for related studies in a slightly different context of diagonals of operators.…”

“…. ., and consult [23] for related studies in a slightly different context of diagonals of operators.…”