The spectrum of shift operators and the existence of traces
Let \(\mathfrak{m}(\mathbb N)\) be a vector space comprising scalar sequences that is endowed with a quasi-norm \(\|\cdot\|_{\mathfrak{m}}\) that makes it complete. (A quasi-norm is a functional \(\|\cdot\|\) on a vector space \(X\) that satisfies all the axioms of a norm except for the triangle inequality, that is replaced by the condition \(\|x+y\| \leqslant C(\|x\|+\|y\|)\) (\(x,y\in X\)) for some universal constant \(C\geqslant 1\).) Moreover, let us assume that \(\mathfrak{m}(\mathbb N)\) is closed under the forward shift operator \(S_+: (x_n)_{n=1}^\infty \mapsto (0,x_1, x_2, x_3, \dots )\) as well as the backward shift operator \(S-+: (x_n)_{n=1}^\infty \mapsto (x_{n+1})_{n=1}^\infty\). The author terms such a structure a quasi-Banach shift-invariant solid sequence space. In the quasi-normed space, it is still meaningful to consider the spectral radius of a (continuous, linear) operator \(S\) acting on a quasi-normed space and the usual formula is in place: \(r(S) = \inf_{n\in\mathbb N} \|S^n\|\), where \(\|S^n\|\) denotes the operator (quasi-)norm defined in terms of the underlying norm in the usual way. Moreover, the spectral radius formula still holds, so that the spectral radius of an operator \(S\) may be computed as the supremum over the moduli of elements of the spectrum of \(S\), \(\sigma(S)\) (in the algebra of continuous linear operators). The author gives a description of the spectrum of the forward shift operator acting on a quasi-Banach shift-invariant solid sequence space by proving the formula that is otherwise familiar in the classical case \[ \sigma(S_+) = \{\lambda: \mathbb C: |\lambda|\leqslant r(S_+), \] thereby providing an answer to a special, yet important case of Problem 7.19 posed in [\textit{A.\,Pietsch}, Integral Equations Oper. Theory 79, No. 2, 255--299 (2014; Zbl 1337.47031)]. Under some mild conditions on the underlying space, the same formula is proved for the backward shift operator \(S_-\), too. (It remains open whether this holds for backward shifts on all quasi-Banach shift-invariant solid sequence spaces.) These general results are then used to study (non-)existence of traces on quasi-Banach ideals of operators on a separable Hilbert space that are determined by a quasi-norm on the diagonal operators belonging therein. In particular, the author translates a theorem concerning the Boyd index, proved in [\textit{K.\,Dykema} et al., Adv. Math. 185, No. 1, 1--79 (2004; Zbl 1103.47054)], to a statement related to the spectral radius formula.
scientific article; zbMATH DE number 6882304
scientific article