We study ℓ _∞ -valued and ℓ _1 -valued noncommutative symmetric spaces. We define related Hardy spaces of noncommutative martingales, show duality and interpolation properties of these spaces. Using these results, we give a direct proof of algebraic atomic decompositions and Davis-type decompositions within the context of Hardy spaces affiliated with separable noncommutative symmetric spaces, which constitute an interpolation of the (L_p, L_q) pair for 1< p ≤ q < 2 . We also derive a version of Doob maximal inequality associated with symmetric spaces of measurable operators.
An operator T on ℋ is called pseudo-selfadjoint with S if T is densely defined and there exists a boundedly invertible operator S on ℋ such that T^* = STS^-1 . In this paper, we investigate the distance from a pseudo-selfadjoint operator to the set of selfadjoint operators. We also study some conditions for an operator to be selfadjoint when powers of such an operator are pseudo-selfadjoint. Furthermore, we investigate various local spectral properties of pseudo-selfadjoint operators, including hypercyclicity and weak hypercyclicity. In addition, we show that T is an invertible subnormal operator if and only if T admits a Hermitian factorization of the form T=AB for two selfadjoint operators A and B.
This paper studies the topological (parameter) entropy of two-parameters systems, as defined by Tsukamoto [39]. We first prove that this entropy is invariant under topological conjugacy and discuss several of its basic dynamical properties. We then compare it with the classical topological entropy of Adler et al. [1] and the extended topological entropy introduced by Cheng [11]. Finally, we define a measure-theoretic parameter entropy via a common invariant measure and establish a variational inequality linking it to the topological (parameter) entropy of two-parameters systems.
For fractional discrete nonautonomous p-Laplace equations driven by superlinear noise, we construct a pullback measure attractor in the space of all probability measures on Banach space l^p (which is larger than l^2 ) for p>2 . Using a so-called two-stage approach which is different from the recent ten references for other models in Hilbert spaces, we provide the estimates for exponential decay moments and tail moments in the Banach space, which lead to the pullback absorption and asymptotic tightness of the dual measure process. Moreover, we prove the upper semicontinuity of the enlarged measure attractors with respect to the noise intensity. It is the first time to study measure attractors when the underlying space is a Banach space.
Using Calderón–Zygmund techniques, we establish a link between gauge-controlled mean oscillation bounds and Hölder-type continuity. We introduce a gauge-based notion of Lebesgue set and examine its measurability, together with its approximation properties. For power gauges, we prove rectifiability results showing that, up to negligible sets, the graphs of functions on their Lebesgue sets can be covered by countably many Hölder graphs.
We study structural properties of the free Banach lattice FBL⟨ L⟩ generated by a distributive lattice L. We characterize when FBL⟨ L⟩ has a strong unit, compute its density character, analyze the density character of order intervals and study when is FVL⟨ L⟩ order dense in FBL⟨ L⟩. We also study projection bands, quasi-interior points, and Banach lattice homomorphisms induced by lattice homomorphisms. Finally, we show that FBL⟨ L⟩ is lattice isometric to FBL⟨ L^op⟩, where L^op denotes the opposite lattice.
A. Beurling used the tools of functional analysis to solve the problem of characterizing the shift-invariant subspaces of the Hardy space H^2, thus establishing the Beurling theorem. In this paper, we extend the Beurling invariant subspace theorem to the weighted Lebesgue and Hardy spaces for a larger class of symmetric gauge norms (e.g., Orlicz, Lorentz, Marcinkiewicz norms), beyond classical L^p -norms. A crucial tool employed in the proof of our main result is a new density theorem for the Lebesgue spaces L^α _ω (𝕋). This theorem serves as a bridge connecting the invariant subspaces under the weak*-topology and the α _ω -topology. As an application, we characterize the outer functions as cyclic vectors and discuss the inner–outer factorization theorem on the weighted Hardy spaces H^α _ω (𝕋).
This paper is devoted to the study of topological pressure for dynamical systems with local time, including both classical and nonlinear cases. For the classical topological pressure for dynamical systems with local time, we investigate equilibrium states, subdifferential and freezing states, while also discussing some basic properties of it. Additionally, the high dimensional nonlinear topological pressure for dynamical systems with local time is introduced, and the corresponding variational principle is established.
We continue the study by the first author on isometric and unitary Toeplitz + Hankel operators (Gu in J Math Anal Appl 552:129797, 2025). We give a complete description of the symbols of unitary (T+H)-operators. By factorizing a large class of isometric (T+H)-operators as products of Toeplitz operators and unitary (T+H)-operators, we find the spectrum of these isometries. Explicit examples of isometric and unitary (T+H)-operators are presented to illustrate that these operators are interesting and are worthy of further investigation.
In this paper, we focus on the problem of phase retrieval from intensity measurements of the short-time linear canonical transform (STLCT). Specifically, we show that the STLCT enables unique recovery of any square-integrable function in high-dimensional space L^2(ℝ^d) through phaseless STLCT sampling on rectangular square-root lattices. When considering the uniform lattices, we construct counterexamples in L^2(ℝ) that demonstrate the limitations of STLCT phase retrieval in this setting. Nevertheless, for functions in band-limited function spaces, phase retrieval can still be achieved on uniform lattices.
Due to Walsh functions being a discontinuous alternative to trigonometric functions, in contrast to usual trigonometric function-based Gabor analysis, the Walsh function-based Gabor analysis has potential applications in processing discontinuous signals. This paper addresses such Gabor analysis in the setting of periodic subspace L^2(S) with S⊆ℝ_+=[0, ∞ ) and S⊕αℤ_+=S , where α >0 and “ ⊕ " is a kind of Walsh addition defined on ℝ_+ . Using “ ⊕ "-based Zak transform, we characterize the complete condition of Gabor systems, Gabor frames (Riesz bases, orthonormal bases) and (weak) Gabor duals in L^2(S) . Also some examples have been provided.
Existence of strong solutions for a Dirichlet problem driven by a Duffing-type differential inclusion is achieved in the setting of separable reflexive Banach spaces. This general framework makes it possible to model and analyse complex phenomena such as periodic or chaotic dynamics as well as oscillatory behavior. The method used to obtain the existence results is based on the combination of a fixed point theorem and a selection theorem. By using the weak topology we avoid assumptions of compactness on the multivalued term. Our abstract results allow to deduce the existence of an admissible pair for a control problem driven by a Duffing-type differential equation. The paper ends with the study of optimal control problem involving a suitable functional.
The nonlinear Moore–Penrose metric generalized inverse was introduced to analyze the best approximate solutions to ill-posed operator equations in Banach spaces. In this paper, under suitable conditions, utilizing certain geometric properties of Banach spaces and features of the metric projection, we develop some iterative methods for computing the Moore–Penrose metric generalized inverse of Banach space operators, deriving some conditions for iterative convergence. Furthermore, we provide error bounds for the iterative methods approximating the Moore–Penrose metric generalized inverse. As a supplementary component to our main findings, an iterative method for deriving the best approximate solution to ill-posed operator equations is proposed. Several relevant examples are also provided to illustrate the efficacy and application of these iterative methods. The results presented in this paper provide new insights into the analysis, computation, and applications of nonlinear generalized inverses.
We study iterated weighted residual (WR) splittings generated by a positive operator R_0∈ B(H)_+ and a finite family of contractions C_1,…,C_m in B(H). The associated residual update R↦ R^1/2(I-C^*_jC_j)R^1/2 produces an m-ary energy tree of residuals { R_w} and dissipated pieces { D_w,j} indexed by finite words. From this tree we construct intrinsic path measures on the path space by biasing transitions either by a fixed quadratic form x↦⟨ x,D_w,jx⟩ (defining the measures ν_x) or, in the trace-class setting, by tr(D_w,j) (yielding a reference measure ν_tr). When R_0∈ S_1(H)_+, we show that ν_tr dominates the family { ν_x} and identify dν_x/dν_tr as a canonical martingale limit of cylinder likelihood ratios. Along ν_tr-almost every branch the residuals decrease to a terminal trace-class random variable R_∞, which we interpret as the WR boundary variable. We then disintegrate ν_tr over σ(R_∞), obtaining a boundary law μ_tr=(R_∞)_#ν_tr and conditional path measures { ν^T_tr}. Finally, we show that each ν_x admits a boundary representation as a mixture of { ν^T_tr} with an explicit boundary density h_x=dμ_x/dμ_tr, thereby organizing the family of intrinsic WR path measures by a single trace-biased boundary disintegration.
Let ℱ_e be a Parseval frame in a Hilbert space ℋ and let be a set of real numbers. From these data, we construct an operator H_,e and a positive operator-valued measure (POVM) F_,e This paper investigates in detail the relationship between the operator H_,e and the POVM F_,e . Our results extend the classical correspondence between a self-adjoint operator generated by an orthonormal basis and its associated projection-valued (spectral) measure.
In this paper, we study the asymptotic behavior of singular values of compact Toeplitz and Hankel operators on Fock-type spaces F^2_Ψ over ℂ^d . For Toeplitz operators T_μ induced by positive Borel measure μ , we characterize the asymptotic behavior of the singular values sequence {s_n(T_μ )}_n=1^∞ in terms of the non-increasing rearrangement of the local averaging function and the Berezin transform of μ . For Hankel operators H_f with locally integrable symbols, we establish asymptotic estimates for {s_n(H_f)}_n=1^∞ via the non-increasing rearrangement of a certain Luecking-type function.
We aim to investigate the dimension theory of α -pressure-like quantities. By means of the Carath é odory-Pesin structure, we define α -BS dimension and α -Pesin topological pressure on subsets using α -Bowen metric d_n^α(x,y)=max _0≤ i≤ n-1e^α id(f^ix,f^iy), where α≥ 0 . Specifically, we show that α -BS dimension and α -Pesin topological pressure are related by a Bowen’s equation. Inspired by the classical Brin–Katok entropy, we introduce the notion of α -local Brin–Katok entropy, and establish a variational principle for α -BS dimension on compact subsets in terms of α -local Brin–Katok entropy. Besides, for subshifts of finite type, we prove that α -Bowen topological entropy is closely related to spectral radius and Hausdorff dimension.
A quasi-projection pair consists of two operators P and Q acting on a Hilbert C^* -module H, where P is a projection and Q is an idempotent satisfying Q^*=(2P-I)Q(2P-I) , in which Q^* denotes the adjoint operator of Q, and I is the identity operator on H. Such a pair is said to be harmonious if both P(I-Q) and (I-P)Q admit polar decompositions. The primary goal of this paper is to present block matrix representations for a harmonious quasi-projection pair (P, Q) on a Hilbert C^* -module, and additionally to derive new block matrix representations for the matched projection, range projection, and null space projection of Q. Several applications of these newly obtained block matrix representations are also explored.
In 1978, Cowen and Douglas introduced a class of geometric operators (known as Cowen–Douglas class of operators) and associated a Hermitian holomorphic vector bundle to such operators. They gave a complete set of unitary invariants in terms of the curves and its covariant derivatives. In this paper, after giving some basic properties of S-spectrum and right eigenvalues of bounded right linear operators on separable quaternionic Hilbert spaces, we generalize the class of Cowen–Douglas operators to the quaternionic Hilbert space via the S-spectrum and denote this class as B_n^s(Ω _q) . Due to the lack of commutativity of quaternion multiplication, the quaternionic Cowen–Douglas operators are not trivial generalizations of the classical Cowen–Douglas operators. Each operator in B_n^s(Ω _q) corresponds to an n-dimensional Hermitian right holomorphic quaternionic vector bundle. We first establish a rigidity theorem for Hermitian right holomorphic quaternionic vector bundles. It is then proven that two operators in B_n^s(Ω _q) are quaternion unitarily equivalent if and only if the associate bundles are equivalent as Hermitian right holomorphic quaternionic vector bundles. In particular, we introduce canonical matrix representations of operators in B_1^s(Ω _q) and furthermore, we give the quaternion unitarily equivalent classification of B_1^s(Ω _q) by the canonical matrix representations. It is worth noting that curvature is a complete unitary invariant for the classical (complex) Cowen–Douglas operators, however, there exist two quaternionic Cowen–Douglas operators which have the same curvature but are not quaternion unitarily equivalent. In addition, we prove that the operators in B_1^s(Ω _q) are quaternion unitarily equivalent if and only if their complex representations are unitarily equivalent. Some relevant examples of the above results are also provided.
