This paper studies the relationships between the Hausdorff measure of noncompactness and various operator seminorms on Banach spaces. We establish sharp two-sided estimates for the Hausdorff measure of noncompactness of operators and their extensions to superspaces, and extend such estimates to the asymptotic noncompactness of C_0 -semigroups. It is proved that the Hausdorff measure of an operator coincides with a naturally defined operator seminorm, and equals the essential norm on Banach spaces admitting a monotone or weakly monotone Schauder basis. We further characterize the asymptotic behavior and iterative convergence of noncompactness indices of C_0 -semigroups, and establish ordered inequalities connecting the Hausdorff measure, essential norm and approximation numbers. The results enrich the quantitative theory of measures of noncompactness and offer new tools for studying noncompact operators and semigroups on Banach spaces.
We investigate several normality-type properties of bounded Volterra-type integral operators V_g acting on the Fock space. We show that this class contains no nonzero normal or quasinormal operators, and that V_g is binormal if and only if either the quadratic or the linear term in the polynomial representation of the symbol g vanishes. We further prove that the properties of being normaloid, subnormal, hyponormal, paranormal, totally paranormal, p-hyponormal, * -paranormal, and quasi- * -paranormal are all equivalent and are characterized by the absence of the linear term in g. These results reveal a clear distinction between the roles of the linear and quadratic terms in determining the normality behavior of these operators. Finally, we show that every bounded Volterra-type integral operator on the Fock space is essentially normal.
We construct a self-adjoint Laplacian on a heterogeneous medium coupling a bounded Lipschitz open set Ω⊂ℝ^n to a finite weighted graph G=(V,E) , through the closed symmetric bilinear form ℰ_ε (u,u) = ‖∇ u_c‖ _L^2(Ω )^2 + ∑ _i∈ Vκ _i |(A_ε u_c)(i)-u_d(i)|^2 + ∑ _{i,j}∈ Ew_ij |u_d(i)-u_d(j)|^2, where A_ε averages u_c over a ball of radius ε at each coupling point; Kato’s representation theorem supplies the generator L_ε . We study the limit ε↓ 0 . In dimension n=1 the coupling survives: we prove Mosco, norm resolvent and spectral convergence with multiplicities, for finite coupling strengths and in the constrained regime κ _i→ +∞ . In every dimension n≥ 2 it does not: for any coupling strengths whatsoever, including the capacity rescalings suggested by point-interaction heuristics, the forms converge to the fully decoupled one, singletons having zero H^1 -capacity, so that the coupling constraint costs asymptotically no energy. The dichotomy governing the model is thus n=1 against n≥ 2 , not the n≤ 3 against n≥ 4 of point-interaction theory. An analytical study and three finite-element experiments illustrate both regimes.
Let X be an infinite-dimensional complex Banach space and let L(X) denote the algebra of all bounded linear operators on X. In this paper, we study the spectra arising from Fredholm theory and Weyl-type theorems of an operator T ∈ L(X) that commutes with a quasinilpotent operator Q ∈ L(X) satisfying α (Q)<∞ and p(Q)<∞ , where α (Q) and p(Q) denote the nullity and the ascent of Q, respectively.
In this paper, we study the dynamical properties of weighted backward shifts B_λ on the little generalized Hardy space on a rooted tree. More precisely, we provide a characterization of ℱ -transitivity for B_λ . In addition, we characterize weighted backward shifts on rooted directed trees that have an orbit with a nonzero limit point. Using these equivalent conditions, we construct several concrete examples illustrating different dynamical behaviors of weighted backward shifts.
