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One of our main results is the following: Let$X$be a compact connected subset of the Euclidean space$R$^{$n$}and$r(X, d$_{2}) the rendezvous number of$X$, where$d$_{2}denotes the Euclidean distance in$R$^{$n$}. (The rendezvous number$r(X, d$_{2}) is the unique positive real number with the property that for each positive integer$n$and for all (not necessarily distinct)$x$_{1},$x$_{2},...,$x$_{$n$}in$X$, there exists some$x$in$X$such that $$(1/n)\sum\nolimits_{i = 1}^n {d_2 (x_i ,x)} = r(X,d_2 )$$ .) Then there exists some regular Borel probability measure μ_{0}on$X$such that the value of ∫_{$X$}$d$_{2}($x, y$)$d$μ_{0}($y$) is independent of the choice$x$in$X$, if and only if$r$($X, d$_{2}) = sup_{μ}∫_{$X$}∫_{$X$}$d$_{2}($x, y$)$d$μ($x$)$d$μ($y$), where the supremum is taken over all regular Borel probability measures μ on$X$.

Nonself-adjoint, non-dissipative perturbations of possibly unbounded self-adjoint operators with real purely singular spectrum are considered under an additional assumption that the characteristic function of the operator possesses a scalar multiple. Using a functional model of a nonself-adjoint operator (a generalization of a Sz.-Nagy–Foiaş model for dissipative operators) as a principle tool, spectral properties of such operators are investigated. A class of operators with almost Hermitian spectrum (the latter being a part of the real singular spectrum) is characterized in terms of existence of the so-called weak outer annihilator which generalizes the classical Cayley identity to the case of nonself-adjoint operators in Hilbert space. A similar result is proved in the self-adjoint case, characterizing the condition of absence of the absolutely continuous spectral subspace in terms of the existence of weak outer annihilation. An application to the rank-one nonself-adjoint Friedrichs model is given.

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