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.
Let A be a unital not necessarily commutative C*-algebra. In this manuscript, we define that a bounded adjointable operator between two Hilbert A-modules is a Fredholm operator if it has finitely generated projective kernel and cokernel as a straight forward generalization of the classical definition. We generalize the classical Atkinson theorem and a version of Atiyah–Janich theorem to show that the index holds the homotopy invariance property for this class of operators.
We obtain some conditions under which all the eigenvalues of a matrix lie at the vertices of a regular polygon. We also consider the case when the eigenvalues lie at the vertices of two concyclic regular polygons. The special cases yield some interesting results about the eigenvalues of unitary matrices and their geometric configurations. Some results related to the similarity of two unitary matrices are also given here.
We study mappings T from a unital C^* -algebra A into its dual space, A^* , which satisfy the ternary derivation identity with respect to the quadratic Jordan product. We prove that whenever A admits a projection p that is Murray–von Neumann equivalent to 1-p , such mappings are automatically additive and conjugate homogeneous over ℚ+iℚ , and they become ternary derivations when they are continuous.
Using operator inequalities, we study a Wold-type decomposition of covariant representations. Building on this decomposition, we prove a Beurling-type theorem showing that every nonzero invariant subspace is uniquely determined by its wandering subspace. Our results extend classical theorems of Beurling and subsequent developments for left-invertible operators to the setting of covariant representations of C^*-correspondences, providing a unified framework for invariant subspace theory under operator inequalities.
Proceeding through a representation theoretic approach, we study the family of Toeplitz operators on the weighted Bergman space ℋ^2_α(𝔹_n) whose symbols are invariant under the action of the semidirect group S_Δ⋊𝕋^n , where S_Δ is a subgroup of permutations. Associating a suitable partition Δ we associated the conjugation J_A on the Bergman space. Moreover we characterize the complex symmetric Toeplitz operators T_a and T_a(r)t^pt^q with respect the conjugation J_A , where the symbols are invariant under the action of S_Δ⋊𝕋^n .
In this paper, we establish a necessary and sufficient condition for a Toeplitz operator with a bounded pluriharmonic symbol to commute with a Hankel operator with a bounded symbol on the Hardy space of the bidisk.
This paper is a sequel of a recent work regarding an open problem posted by Lau and Zhang on a characterization of existence of a left invariant mean on WAP(S), the Banach algebra of weakly almost periodic functions of a semitopological semigroup S, in terms of a fixed point property for nonexpnsive mappings on weakly compact convex sets in locally convex spaces. More precisely, it is about whether (left) amenability of WAP(S) of a given semitopological semigroup S is equivalent to the following fixed point property: (F): Every weakly separately continuous, weakly quasi-equicontinuous and Q-nonexpansive action of S on a weakly compact convex set K in a separated locally convex topological vector space (E, Q) has a common fixed point for S. In this paper, we provide a complete answer to this question for arbitrary semitopological semigroups.
Let ℬ(ℋ) be the algebra of all bounded linear operators on an infinite-dimensional Hilbert space ℋ . A map Δ , from ℬ(ℋ) into a closed subset of ℂ is said to be a ∂ -spectrum if ∂σ (T) ⊆Δ (T) ⊆σ (T) for all T ∈ℬ(ℋ) , where σ (T) denotes the spectrum of T and ∂σ (T) its boundary. Fix a ∂ -spectrum map Δ . In this paper, we characterize all maps ϕ :ℬ(ℋ) →ℬ(ℋ) whose ranges contain all operators of rank at most two and that satisfy either Δ (TS^*) = Δ (ϕ (T)ϕ (S)^* ) for all T,S ∈ℬ(ℋ) or Δ (TS^*T) = Δ (ϕ (T)ϕ (S)^*ϕ (T) ) for all T,S ∈ℬ(ℋ).
We prove Grothendieck–Pietsch composition results for multiple summing operators without a linear analogue. As application, we give a new proof for the multilinear version of Grothendieck’s composition theorem proven by D. P érez-García and I. Villanueva. We use this result in tandem with the Maurey factorization theorem, to find the necessary and sufficient conditions for multilinear multiplication operator from a product of l_p spaces into l_1 to be multiple 2-summing.
For any closed K⊆ℝ^n , recently all K-positivity preserver have been characterized, i.e., all linear operators T:ℝ[x_1,… ,x_n]→ℝ[x_1,… ,x_n] such that Tp≥ 0 on K for all p≥ 0 on K. An important extension of polynomials ℝ[x_1,… ,x_n] with real coefficients are polynomials ℝ^m× m[x_1,… ,x_n] with matrix coefficients. Non-negativity on K for matrix polynomials with Hermitian coefficients Herm_m is then p(x)≽ 0 for all x∈ K . In the current work, we investigate linear operators T:Herm_m[x_1,… ,x_n]→Herm_m[x_1,… ,x_n] . We focus on matrix K-positivity preserver, i.e., Tp≽ 0 on K for all p≽ 0 on K. For K=ℝ^n and compact sets K⊆ℝ^n , we give characterizations of matrix K-positivity preservers. We discuss the difference between the real and the matrix coefficient case and where our proof fails for general sets K⊆ℝ^n with Kℝ^n and K non-compact.
We prove a Fredholm criteria for singular integral operators with continuous coefficients on any separable rearrangement-invariant Banach function space X(ℝ) with nontrivial Boyd indices, and variable Lebesgue spaces with p(· )∈ℬ_M(ℝ) .
We study a class of nonlocal Kirchhoff-type problems involving the fractional p -Laplacian and critical Sobolev growth. The equation includes a Kirchhoff term M(t) = a + t^m , with a ≥ 0 and m > 0 , and is posed on a bounded domain with Lipschitz boundary. Using variational methods, Krasnoselskii’s genus theory, and a fractional concentration-compactness principle, we prove the existence of infinitely many weak solutions in both the non-degenerate ( a > 0 ) and degenerate ( a = 0 ) cases.
Given a weight v on an open set G of a Banach space E, the weighted spaces of analytic functions Hv(G) and Hv_0(G) and their interpolating sequences are studied. In particular, it is shown that for a large class of weights on B_E we have that Hv(G) is naturally isometrically isomorphic to Hv_0(G)^** , and that for those weights the interpolating sequences for Hv(B_E) are completely classified by their boundary behavior either as the interpolating sequences for Hv_0(B_E) or for the classical Hardy space H^∞ (B_E) .
Suppose Q is an idempotent operator. Let Q=V_Q|Q| and Ψ =U_Ψ |Ψ | be the polar decompositions of Q and Ψ =2Q-I , respectively. Let Φ =Q+Q^* and Υ =Q+Q^*-I . We prove that Q=|Q^*||Q|=1/2 [(2|Φ |-|Ψ |)U_Ψ -I ], |Φ |=|Q|+|Q^*|= |Υ |+ U_Ψ, |Q|-|Q^*|=1/2(|Ψ |-|Ψ ^*|)=|Ψ |-|Υ |=|Υ |-|Ψ ^*| and U_Ψ= U^*_Ψ= U^-1_Ψ=|Ψ |Ψ =|Ψ ^*|Ψ ^*=Ψ |Ψ ^*| =Ψ ^*|Ψ |=|Υ |Υ ^-1=|Υ |^-1Υ . The equivalent conditions for positive operators A and B which can be written as A=Q^*Q and B=QQ^* are obtained. Also, we characterize the idempotents Q, Q_1 and Q_2 such that ℛ(Q) ⊆ℛ(Q_1) or 𝒩(Q_2) ⊆𝒩(Q) . In particular, the equivalent condition for idempotent Q which can be written as the product Q=Q_1Q_2 is described.
Let (K,H_V) be a Heisenberg Gelfand pair and G:=K⋉ H_V be its associated semidirect product. Here, K is a compact Lie group acting smoothly on the Heisenberg group H_V:=V×ℝ, where V is a finite-dimensional complex vector space. Let G be the unitary dual of G equipped with the Fell topology. We say that π∈G is a spherical representation of G if the restriction π |_K of π to the subgroup K has a one-dimensional space of K-fixed vectors. Boidol, Ludwig and Müller have introduced the notion of the so-called small representations of G, that is all π∈G that cannot be Hausdorff separated from the trivial one-dimensional representation 1_G of G. Using a parametrization of the set of irreducible spherical representations due to work of Benson-Ratcliff, we show that any non-trivial spherical representation of G cannot be a small representation. Furthermore, we prove that the converse of this statement is false in the setting of the Heisenberg motion group G_d:=U(d)⋉ H_ℂ^d, d∈ℕ^× .
Frame theory has important applications in phase-retrieval problem, and the dimension function of a frame can serve as a candidate for measuring its phase retrievability. In this paper, we first investigate the relationship between phase retrievability and the dimension function of g-frames incorporating operator theory. We find that the dimension function of a phase-retrievable g-frame closely aligns with that of an ordinary frame. This consistency suggests that the dimension function of a g-frame can likewise act as a metric for measuring its phase retrieval capability. Specifically, we find that the exact PR-redundancy of a g-frame is equivalent to the exactness of its dimension function. The paper then investigates the structure of the dimension function range of g-frames. We find that, even when the number of operators in a g-frame is not less than n, significant differences remain from the case of ordinary frames regarding the structure of the dimension function range of g-Riesz basis and whether the range contains n. Finally, we present a method for examining the dimension function of a g-frame via its induced frame, along with a straightforward approach to construct g-frames whose dimension function range contains n.
In this paper, we investigate the existence of pullback attractors for a non-autonomous nonclassical diffusion equation with state-dependent delay. First, the existence and uniqueness of strong solutions are proved via the standard Faedo-Galerkin method. To address the state-dependent delay, we incorporate its compensator into a normal functional, which enables us to obtain both the existence of a pullback absorbing set and the pullback asymptotic compactness of the corresponding evolution process. In particular, new properties of specific composition operators are derived to support the main results. Finally, we establish the existence of pullback attractors.
