A structured inverse spectrum problem for infinite graphs

A Structured Inverse Spectrum Problem for Infinite Graphs And Unbounded Operators

“…Proof. First, observe that Dom(T ) contains all finite sequences and hence it is dense in 2 . Next, since {λ n } ∞ n=1 is a sequence of real numbers, T is clearly symmetric and thus T * is an extension of T .…”

“…is a bounded functional. This functional has a unique bounded extension to 2 and, therefore, by the Riesz representation theorem, it can be represented by a unique c ∈ 2 . That is,…”

“…The solution of the example above illustrates the construction used in [2] to prove the following theorem.…”

“…This inverse spectrum problem was considered in [2] under the assumption that the given spectrum Λ is compact. Since the spectrum of any bounded operator is a compact subset of the complex plane, the main result of [2] is in this sense optimal for bounded operators.…”

“…This inverse spectrum problem was considered in [2] under the assumption that the given spectrum Λ is compact. Since the spectrum of any bounded operator is a compact subset of the complex plane, the main result of [2] is in this sense optimal for bounded operators. Additionally, because the spectrum of any unbounded operator is a closed subset of the complex plane (see, for instance, [5,Proposition 2.6]), it is natural to ask whether one can replace the compactness assumption of Λ by closedness.…”