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.
In this paper, we explore the classes of n-EP and n-HEP operators, which generalize EP and HEP operators. We investigate their fundamental properties, characterizations, and interrelationships with classical concepts like normality and the Cauchy dual. Our main results include conditions under which these operators coincide and characterizations via range inclusions and commutator identities involving the Moore–Penrose inverse.
Let m=(m_1,m_2,… ) be a bounded generating sequence, 2≤ m_n≤Λ <∞ , and let {ψ _j}_j≥ 0 be the associated Vilenkin system. For every fixed q>2 , we construct a formal Vilenkin series with coefficients in ℓ ^q whose ordinary partial sums have a uniform exceptional-set universality property. Precisely, for every ε >0 there exists a measurable set E_ε⊂ [0,1) with mesE_ε <ε , chosen independently of both the exponent p and the target g, such that for every 1≤ p<∞ and every g∈ L^p[0,1) there is a subsequence of the ordinary partial sums, allowed to depend on p and g, which converges to g in L^p([0,1)∖ E_ε ) . The proof is based on a quantitative localization lemma: from an m-adic step function and a prescribed initial spectral index we construct a high-frequency Vilenkin polynomial which reproduces the function outside a small set and has arbitrarily small ℓ ^q coefficient mass. The required multiplier is selected probabilistically on a finite abelian digit group. We also distinguish two endpoint phenomena at q=2 : Parseval’s identity obstructs the particular finite-block multiplier mechanism in ℓ ^2 , whereas the Riesz–Fischer theorem rules out the stated global universality property for every ℓ ^2 coefficient sequence.
K–g-fusion frame is a generalization of several currently popular frames, including the fusion frames, K-fusion frames and g-fusion frames. And structured frames are one of most interest current research subjects. In this paper, we mainly study structured K–g-fusion frames, i.e. K–g-fusion frames generated by an operator-space pair through a unitary system. We call such pair a K–g-fusion frame generator of the unitary system. We study some properties of K–g-fusion frame generators, which shows that under some condition we can obtain K–g-fusion frame generators by some simple ways. Moreover, under some condition we characterize K–g-fusion frame generators in terms of complete wandering subspaces and generalized local commutants. Finally, we discuss the duals of a K–g-fusion frame generator, which is called K–g-fusion dual Bessel generator of the generator. And under some weak condition, we completely characterize the structure of all K–g-fusion dual Bessel generators for a Parseval K–g-fusion frame generator and give an explicit construction of the canonical K–g-fusion dual Bessel generator for a Parseval K–g-fusion frame generator. Moreover, we show that under certain condition, existence of dual on range of K guarantees a K–g-fusion Bessel generator to be a K–g-fusion frame generator, and we also can generate a K–g-fusion frame generator from a K–g-fusion dual Bessel generator.
