Weighted Join Operators on Directed Trees
نویسندگان
چکیده
A rooted directed tree $${\mathscr {T}}=(V, E)$$ with root $${\textsf{root}}$$ can be extended to a graph $$\mathscr {T}_\infty =(V_\infty , E_\infty )$$ by adding vertex $$\infty $$ V and declaring each in as parent of . One may associate the {T}}_{\infty }$$ family semigroup structures $$\sqcup _{\mathfrak b}$$ extreme ends being induced join operation meet $$\sqcap from lattice theory (corresponding $$\mathfrak b={\textsf{root}}$$ b= \infty respectively). Each structure among these leads densely defined linear operators $$W^{(\mathfrak b)}_{\varvec{\lambda }_{u}}$$ acting on $$\ell ^2(V),$$ which we refer weighted at given base point b \in V_{\infty prescribed $$u V$$ The this are {{\textsf{root}}})}_{\varvec{\lambda )}_{\varvec{\lambda In paper, systematically study trees. We also present more involved counterpart rootless trees {T}}$$ case, either finite rank operators, diagonal or one perturbations operators. possibly infinite (possibly unbounded) both cases, class overlaps well-studied classes complex Jordan n-symmetric An important half paper is devoted extensions $$W_{f, g}$$ trees, where $$f \ell ^2(V)$$ $$g: \rightarrow {\mathbb {C}}$$ unspecified. Unlike not necessarily closed. provide couple compatibility conditions involving weight system $$\varvec{\lambda }_u$$ g ensure closedness These intimately related whether an associated discrete Hilbert transform well-defined. discuss role Gelfand-triplet realization space adjoint Further, describe various spectral parts terms data. sufficient for sectorial operator (resp. infinitesimal generator quasi-bounded strongly continuous semigroup). case leafless, characterize admit compact resolvent. Motivated above graph-model, take brief look into general non-selfadjoint perturbations.
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ژورنال
عنوان ژورنال: Complex Analysis and Operator Theory
سال: 2023
ISSN: ['1661-8254', '1661-8262']
DOI: https://doi.org/10.1007/s11785-023-01334-y