We strengthen the maximal ergodic theorem for actions of groups of polynomial growth to a form involving jump quantity, which is the sharpest result among the family of variational or maximal ergodic theorems. As two applications, we first obtain the upcrossing inequalities with exponential decay of ergodic averages and then provide an explicit bound on the convergence rate such that the ergodic averages with strongly continuous regular group actions are metastable (or locally stable) on a large interval. Before exploiting the transference techniques, we actually obtain a stronger result—the jump estimates on a metric space with a measure not necessarily doubling. The ideas or techniques involve martingale theory, non-doubling Calderón–Zygmund theory, almost orthogonality argument, and some delicate geometric argument involving the balls and the cubes on a group equipped with a not necessarily doubling measure.

The concept of parity due to Fitzpatrick, Pejsachowicz and Rabier is a central tool in the abstract bifurcation theory of nonlinear Fredholm operators. In this paper, we relate the parity to the Evans function, which is widely used in the stability analysis for traveling wave solutions to evolutionary PDEs.

In contrast, we use Evans function as a flexible tool yielding general sufficient condition for local bifurcations of specific bounded entire solutions to (Carathéodory) differential equations. These bifurcations are intrinsically nonautonomous in the sense that the assumptions implying them cannot be fulfilled for autonomous or periodic temporal forcings. In addition, we demonstrate that Evans functions are strictly related to the dichotomy spectrum and hyperbolicity, which play a crucial role in studying the existence of bounded solutions on the whole real line and therefore the recent field of nonautonomous bifurcation theory. Finally, by means of non-trivial examples we illustrate the applicability of our methods.

We establish higher-order trace formulas for pairs of contractions along a multiplicative path generated by a self-adjoint operator in a Schatten-von Neumann ideal, removing earlier stringent restrictions on the kernel and defect operator of the contractions and significantly enlarging the set of admissible functions. We also derive higher-order trace formulas for maximal dissipative operators under relaxed assumptions and new simplified trace formulas for unitary and resolvent comparable self-adjoint operators. The respective spectral shift measures are absolutely continuous and, in the case of contractions, the set of admissible functions for the nth-order trace formula on the unit circle includes the Besov class $B^n_{\infty , 1}(\mathbb {T})$. Both aforementioned properties are new in the mentioned generality.

Averaged operators are important in Convex Analysis and Optimization Algorithms. In this article, we propose classifications of averaged operators, firmly nonexpansive operators, and proximal operators using the Bauschke–Bendit–Moursi modulus of averagedness. We show that if an operator is averaged with a constant less than $1/2$, then it is a bi-Lipschitz homeomorphism. Amazingly the proximal operator of a convex function has its modulus of averagedness less than $1/2$ if and only if the function is Lipschitz smooth. Some results on the averagedness of operator compositions are obtained. Explicit formulae for calculating the modulus of averagedness of resolvents and proximal operators in terms of various values associated with the maximally monotone operator or subdifferential are also given. Examples are provided to illustrate our results.

Recently, the Kac formula for the conditional expectation of the first recurrence time of a conditionally ergodic conditional expectation preserving system was established in the measure-free setting of vector lattices (Riesz spaces). We now give a formulation of the Kakutani–Rokhlin decomposition for conditionally ergodic systems in terms of components of weak order units in a vector lattice. In addition, we prove that every aperiodic conditional expectation preserving system can be approximated by a periodic system.

We study the behaviour of the norm of the resolvent for non-self-adjoint operators of the form $A := -\partial_x + W(x)$, with $W(x) \ge 0$, defined in ${L^2}({\mathbb{R}})$. We provide a sharp estimate for the norm of its resolvent operator, $\| (A - \lambda)^{-1} \|$, as the spectral parameter diverges $(\lambda \to +\infty)$. Furthermore, we describe the C0-semigroup generated by −A and determine its norm. Finally, we discuss the applications of the results to the asymptotic description of pseudospectra of Schrödinger and damped wave operators, and also the optimality of abstract resolvent bounds based on Carleman-type estimates.

Two frames $\{f_n\}_{n =1}^{\infty }$ and $\{g_n\}_{n =1}^{\infty }$ in a separable Hilbert space ${\mathcal H}$ are said to be weaving frames, if for every $\sigma \subset \mathbb N$, $\{f_n\}_{n\in \sigma } \cup \{g_n\}_{n\in \sigma ^c}$ is a frame for ${\mathcal H}$. Weaving frames are proved to be very useful in many areas, such as, distributed processing, wireless sensor networks, packet encoding, and many more. Inspired by the work of Bemrose et al. [2], this paper delves into the properties and characterizations of weaving frames and weaving Riesz bases.

We study density properties of orbits for a hypercyclic operator T on a separable Banach space X, and show that exactly one of the following four cases holds: (1) every vector in X is asymptotic to zero with density one; (2) generic vectors in X are distributionally irregular of type $1$; (3) generic vectors in X are distributionally irregular of type $2\frac {1}{2}$ and no hypercyclic vector is distributionally irregular of type $1$; (4) every hypercyclic vector in X is divergent to infinity with density one. We also present some examples concerned with weighted backward shifts on $\ell ^p$ to show that all the above four cases can occur. Furthermore, we show that similar results hold for $C_0$-semigroups.

We define the Schur–Agler class in infinite variables to consist of functions whose restrictions to finite-dimensional polydisks belong to the Schur–Agler class. We show that a natural generalization of an Agler decomposition holds and the functions possess transfer function realizations that allow us to extend the functions to the unit ball of $\ell ^\infty $. We also give a Pick interpolation type theorem which displays a subtle difference with finitely many variables. Finally, we make a brief connection to Dirichlet series derived from the Schur–Agler class in infinite variables via the Bohr correspondence.

We study the class of composition operators acting on the Fréchet space $\mathrm {Hol}(\mathbb {B}_N)$ of all holomorphic maps on the unit ball of $\mathbb {C}^N$. We describe the conditions to make these operators continuous, invertible and compact. We also do the spectral study of these operators, depending on the nature of its symbol.

We introduce and study the notion of hereditary frequent hypercyclicity, which is a reinforcement of the well-known concept of frequent hypercyclicity. This notion is useful for the study of the dynamical properties of direct sums of operators; in particular, a basic observation is that the direct sum of a hereditarily frequently hypercyclic operator with any frequently hypercyclic operator is frequently hypercyclic. Among other results, we show that operators satisfying the frequent hypercyclicity criterion are hereditarily frequently hypercyclic, as well as a large class of operators whose unimodular eigenvectors are spanning with respect to the Lebesgue measure. However, we exhibit two frequently hypercyclic weighted shifts $B_w,B_{w'}$ on $c_0(\mathbb {Z}_+)$ whose direct sum ${B_w\oplus B_{w'}}$ is not $\mathcal {U}$-frequently hypercyclic (so that neither of them is hereditarily frequently hypercyclic), and we construct a C-type operator on $\ell _p(\mathbb {Z}_+)$, $1\le p<\infty $, which is frequently hypercyclic but not hereditarily frequently hypercyclic. We also solve several problems concerning disjoint frequent hypercyclicity: we show that for every $N\in \mathbb {N}$, any disjoint frequently hypercyclic N-tuple of operators $(T_1,\ldots ,T_N)$ can be extended to a disjoint frequently hypercyclic $(N+1)$-tuple $(T_1,\ldots ,T_N, T_{N+1})$ as soon as the underlying space supports a hereditarily frequently hypercyclic operator; we construct a disjoint frequently hypercyclic pair which is not densely disjoint hypercyclic; and we show that the pair $(D,\tau _a)$ is disjoint frequently hypercyclic, where D is the derivation operator acting on the space of entire functions and $\tau _a$ is the operator of translation by $a\in \mathbb {C}\setminus \{ 0\}$. Part of our results are in fact obtained in the general setting of Furstenberg families.

Chan and Seceleanu have shown that if a weighted shift operator on $\ell^p(\mathbb{Z})$, $1\leq p \lt \infty$, admits an orbit with a non-zero limit point then it is hypercyclic. We present a new proof of this result that allows to extend it to very general sequence spaces. In a similar vein, we show that, in many sequence spaces, a weighted shift with a non-zero weakly sequentially recurrent vector has a dense set of such vectors, but an example on $c_0(\mathbb{Z})$ shows that such an operator is not necessarily hypercyclic. On the other hand, we obtain that weakly sequentially hypercyclic weighted shifts are hypercyclic. Chan and Seceleanu have, moreover, shown that if an adjoint multiplication operator on a Bergman space admits an orbit with a non-zero limit point then it is hypercyclic. We extend this result to very general spaces of analytic functions, including the Hardy spaces.

Isoclinic subspaces have been studied for over a century. Quantum error correcting codes were recently shown to define a subclass of families of isoclinic subspaces. The Knill–Laflamme theorem is a seminal result in the theory of quantum error correction, a central topic in quantum information. We show there is a generalized version of the Knill–Laflamme result and conditions that applies to all families of isoclinic subspaces. In the case of quantum stabilizer codes, the expanded conditions are shown to capture logical operators. We apply the general conditions to give a new perspective on a classical subclass of isoclinic subspaces defined by the graphs of anti-commuting unitary operators. We show how the result applies to recently studied mutually unbiased quantum measurements (MUMs), and we give a new construction of such measurements motivated by the approach.

In this article, motivated by the regularity theory of the solutions of doubly nonlinear parabolic partial differential equations, the authors introduce the off-diagonal two-weight version of the parabolic Muckenhoupt class with time lag. Then the authors introduce the uncentered parabolic fractional maximal operator with time lag and characterize its two-weighted boundedness (including the endpoint case) in terms of these weights under an additional mild assumption (which is not necessary for one-weight case). The most novelty of this article exists in that the authors further introduce a new parabolic shaped domain and its corresponding parabolic fractional integral with time lag and, moreover, applying the aforementioned (two-)weighted boundedness of the parabolic fractional maximal operator with time lag, the authors characterize the (two-)weighted boundedness (including the endpoint case) of these parabolic fractional integrals in terms of the off-diagonal (two-weight) parabolic Muckenhoupt class with time lag; as applications, the authors further establish a parabolic weighted Sobolev embedding and a priori estimate for the solution of the heat equation. The key tools to achieve these include the parabolic Calderón–Zygmund-type decomposition, the chaining argument, and the parabolic Welland inequality, which is obtained by making the utmost of the geometrical relation between the parabolic shaped domain and the parabolic rectangle.

The article deals with isometric dilation and commutant lifting for a class of n-tuples $(n\ge 3)$ of commuting contractions. We show that operator tuples in the class dilate to tuples of commuting isometries of BCL type. As a consequence of such an explicit dilation, we show that their von Neumann inequality holds on a one-dimensional variety of the closed unit polydisc. On the basis of such a dilation, we prove a commutant lifting theorem of Sarason’s type by establishing that every commutant can be lifted to the dilation space in a commuting and norm-preserving manner. This further leads us to find yet another class of n-tuples $(n\ge 3)$ of commuting contractions each of which possesses isometric dilation.

We introduce a general class of transport distances $\mathrm {WB}_{\Lambda }$ over the space of positive semi-definite matrix-valued Radon measures $\mathcal {M}(\Omega, \mathbb {S}_+^n)$, called the weighted Wasserstein–Bures distance. Such a distance is defined via a generalised Benamou–Brenier formulation with a weighted action functional and an abstract matricial continuity equation, which leads to a convex optimisation problem. Some recently proposed models, including the Kantorovich–Bures distance and the Wasserstein–Fisher–Rao distance, can naturally fit into ours. We give a complete characterisation of the minimiser and explore the topological and geometrical properties of the space $(\mathcal {M}(\Omega, \mathbb {S}_+^n),\mathrm {WB}_{\Lambda })$. In particular, we show that $(\mathcal {M}(\Omega, \mathbb {S}_+^n),\mathrm {WB}_{\Lambda })$ is a complete geodesic space and exhibits a conic structure.

Let $\mathcal H$ be a complex, separable Hilbert space, and set . When $\dim \, \mathcal H$ is finite, we characterise the set and its norm-closure . In the infinite-dimensional setting, we characterise the intersection of with the set of biquasitriangular operators, and we exhibit an index obstruction to belonging to .

We investigate when the algebraic numerical range is a C-spectral set in a Banach algebra. While providing several counterexamples based on classical ideas as well as combinatorial Banach spaces, we discuss positive results for matrix algebras and provide an absolute constant in the case of complex $2\times 2$-matrices with the induced $1$-norm. Furthermore, we discuss positive results for infinite-dimensional Banach algebras, including the Calkin algebra.

In our paper [‘Linking the boundary and exponential spectra via the restricted topology’, J. Math. Anal. Appl. 454 (2017), 730–745], we defined and used the restricted topology to establish certain connections among the boundary spectrum, the exponential spectrum, the topological boundary of the spectrum and the connected hull of the spectrum, and in [‘The restricted connected hull: filling the hole’, Bull. Aust. Math. Soc. 109 (2024), 388–392], we presented further properties of the restricted connected hull. We now continue our investigation of the restricted boundary. In particular, we prove a number of mapping and regularity-type properties of the restricted boundary. In addition, we use this concept to provide a new characterisation of the Jacobson radical of a Banach algebra and a variation of Harte’s theorem. Finally, we establish spectral continuity results, in particular, in ordered Banach algebras.

Let $\varphi$ be a normal semifinite faithful weight on a von Neumann algebra $A$, let $(\sigma ^\varphi _r)_{r\in {\mathbb R}}$ denote the modular automorphism group of $\varphi$, and let $T\colon A\to A$ be a linear map. We say that $T$ admits an absolute dilation if there exists another von Neumann algebra $M$ equipped with a normal semifinite faithful weight $\psi$, a $w^*$-continuous, unital and weight-preserving $*$-homomorphism $J\colon A\to M$ such that $\sigma ^\psi \circ J=J\circ \sigma ^\varphi$, as well as a weight-preserving $*$-automorphism $U\colon M\to M$ such that $T^k={\mathbb {E}}_JU^kJ$ for all integer $k\geq 0$, where $ {\mathbb {E}}_J\colon M\to A$ is the conditional expectation associated with $J$. Given any locally compact group $G$ and any real valued function $u\in C_b(G)$, we prove that if $u$ induces a unital completely positive Fourier multiplier $M_u\colon VN(G) \to VN(G)$, then $M_u$ admits an absolute dilation. Here, $VN(G)$ is equipped with its Plancherel weight $\varphi _G$. This result had been settled by the first named author in the case when $G$ is unimodular so the salient point in this paper is that $G$ may be nonunimodular, and hence, $\varphi _G$ may not be a trace. The absolute dilation of $M_u$ implies that for any $1\lt p\lt \infty$, the $L^p$-realization of $M_u$ can be dilated into an isometry acting on a noncommutative $L^p$-space. We further prove that if $u$ is valued in $[0,1]$, then the $L^p$-realization of $M_u$ is a Ritt operator with a bounded $H^\infty$-functional calculus.