We study the problem when an n-tuple of self-adjoint operators in an infinite-dimensional separable Hilbert space H with small commutators is close to an n-tuple of commuting self-adjoint operators on $H.$ We give an affirmative answer to the problem when the synthetic-spectrum and the essential synthetic-spectrum are close. Examples are also exhibited that, in general, the answer to the problem when $n\ge 3$ is negative even the associated Fredholm index vanishes. This is an attempt to solve a problem proposed by David Mumford related to quantum theory and measurements.

Let T be a bounded linear operator on a separable Banach space that satisfies geometric properties similar to those of $\ell ^p,\, p>1$. We prove that the smallest and the largest norm of weak cluster points of all maximizing sequences for T can only take the values $0$ or $1$. The three classes of bounded linear operators emerging from the dichotomy of these extremal norm values coincide with the partition, created by considering the norm-attaining property and if the essential norm equals the norm.

We show that, under certain conditions, a strongly continuous semigroup admits an almost surely frequently hypercyclic random vector defined as a stochastic integral in Fréchet spaces with respect to the Brownian motion. Two criteria are given. We will apply the second criterion to three examples: translation semigroups on spaces of integrable functions, the exponential of weighted shifts, and the translation operators on the space of entire functions. This last example, with a stochastic approach, seems to be new in the literature. Some other examples are given.

We study a discrete process on planar convex bodies in which, at each step, a body is replaced by a weighted Minkowski average of itself and its rotation by a fixed angle. Up to translation and uniform scaling, this produces a rigid averaging dynamical system. We give a complete classification of the limit shapes. If the angle is an irrational multiple of $2\pi $, the iterates converge to a disk. If the angle is rational, they converge to the average of finitely many rotated copies of the initial body. We also obtain sharp convergence rates. In the rational case, the decay is uniform and exponential with an explicit constant depending only on the weight and the denominator of the angle. For irrational angles, we prove quantitative rates under a mild number-theoretic condition that holds for almost every angle: low regularity inputs have polynomial decay up to a logarithmic factor, while real analytic inputs have stretched exponential decay. For angles with bounded continued fraction coefficients, we give matching lower bounds along subsequences. These results describe the global attractors of the dynamics and indicate the absence of chaotic behaviour.

We develop several $\ell ^p$-operator norm inequalities for $k\times k$ block matrices defined on the $\ell ^p$-sum of Banach spaces. Using these inequalities, we obtain p-numerical radius and spectral radius bounds for $k\times k$ block matrices. We deduce a p-numerical radius bound for the Kronecker product $A\otimes B$, where $A\in {M}_k(\mathbb {C})$ is a $k\times k$ complex matrix and $B\in \mathcal {L}(\mathbb {H})$ is a bounded linear operator on a complex Hilbert space $\mathbb {H}$. This improves and extends Holbrook’s bound $w(A\otimes B)\leq w(A)\|B\|.$ If $\|A\|_{\ell ^p}$ and $w_p(A)$ denote the $\ell ^p$-operator norm and p-numerical radius of $A\in {M}_k(\mathbb {C})$, respectively, then it is shown that

$$ \begin{align*} \frac{1}{2}\|A\|_p+\mu_p(A) \leq w_p(A), \end{align*} $$

$\mu _p(A)$

$\ell ^p$

$\frac {1}{2}\|A\|_p= w_p(A)$

In the present article, we study the norms for symmetric and antisymmetric tensor products of weighted shift operators. By proving that for $n\geq 2$,

$$ \begin{align*}\|S_{\alpha}^{l_1}\odot\cdots \odot S_{\alpha}^{l_k}\odot S_{\alpha}^{*l_{k+1}}\odot\cdots \odot S_{\alpha}^{*l_{n}}\| =\mathop{\prod}_{i=1}^n\left \| S_{\alpha}^{{l_{i}}}\right\|, \text{ for any} \ (l_1,l_2\ldots l_n)\in\mathbb N^n\end{align*} $$

We introduce the index ${\mathcal N}(\omega _1,\omega _2)$ of a pair of pure states on a unital C*-algebra, which is a generalization of the notion of the index of a pair of projections on a Hilbert space. We then show that the Hall conductance associated with an invertible state $\omega $ of a two-dimensional interacting electronic system which is symmetric under $U(1)$ charge transformation may be written as the index $\mathcal {N}(\omega ,\omega _D)$, where $\omega _D$ is obtained from $\omega $ by inserting a unit of magnetic flux. This exhibits the integrality and continuity properties of the Hall conductance in the context of general topological features of $\mathcal {N}$.

We consider a normal operator $T$ on a Hilbert space $H$. Under various assumptions on the spectrum of $T$, we give bounds for the spectrum of $T+A$ where $A$ is $T$-bounded with relative bound less than 1 but we do not assume that $A$ is symmetric or normal. If the imaginary part of the spectrum of $T$ is bounded, then the spectrum of $T+A$ is contained in the region between two hyperbolas whose asymptotic slope depends on the $T$-bound of $A$. If the spectrum of $T$ is contained in a bisector, then the spectrum of $T+A$ is contained in the area between certain rotated hyperbolas. The case of infinitely many gaps in the spectrum of $T$ is studied. Moreover, we prove a stability result for the essential spectrum of $T+A$. If $A$ is even $p$-subordinate to $T$, then we obtain stronger results for the localisation of the spectrum of $T+A$.

We solve the dry ten Martini problem for the unitary almost-Mathieu operator with Diophantine frequencies in the non-critical regime.

For bounded operators on Kreĭn spaces, isometric or unitary dilations always exist. We prove that any minimal isometric or unitary dilation has a precise geometrical structure. Moreover, a bounded operator T has a unique minimal unitary dilation if and only if T and $T^*$ have unique minimal isometric dilation if and only if T is either contractive or expansive and $T^*$ is either contractive or expansive. Passing to the bi-dimensional case, a minimal unitary extension (in short, m.u.e.) $U=(U_1, U_2)$ is obtained for a pair $V=(V_1, V_2)$ of commuting bounded isometries on a Kreĭn space. There is a link with the one-dimensional case: if U is an m.u.e. for $V,$ then $U_1U_2$ is an m.u.e. for $V_1V_2$. Also, if $(V_1V_2)^*$ is either contractive or expansive, then V has a unique minimal unitary extension. A minimal regular isometric dilation is obtained for a commuting pair $T=(T_1, T_2)$ of bounded operators on a Kreĭn space such that $T_1,\ T_2$ are contractions and T is a bidisc contraction or $T_1,\ T_2$ are expansions and T is a bidisc expansion. The existence of a minimal unitary extension is used to provide a minimal regular unitary dilation for T. Discussions about uniqueness and geometric structure conclude the article.

In this article, we completely describe the numerical range of Toeplitz operators on weighted Bergman spaces with harmonic symbol. We also characterize the numerical range of weighted composition operators on weighted Bergman spaces and classify some sets which are the numerical range of composition operators. We investigate the inclusion of zero in the numerical range, and compute the radius of circular and elliptical disk contained in the numerical range of weighted composition operators on weighted Bergman spaces.

In this article, we study various properties and characterizations of weaving frames in Krein spaces. In support of our findings, several examples and counter examples are provided, illustrating the applicability of the theoretical results. Additionally, we extend the discussion to an important application in probabilistic erasure, highlighting how weaving frames can be used to mitigate data loss in such scenarios.

Let $\Omega\subset\mathbb{R}^N$, $N\geq 1$, be an open bounded connected set. We consider the indefinite weighted eigenvalue problem $-\Delta u =\lambda m u$ in Ω with $\lambda \in \mathbb{R}$, $m\in L^\infty(\Omega)$ and with homogeneous Neumann boundary conditions. We study weak* continuity, convexity and Gâteaux differentiability of the map $m\mapsto1/\lambda_1(m)$, where $\lambda_1(m)$ is the principal eigenvalue. Then, denoting by $\mathcal{G}(m_0)$ the class of rearrangements of a fixed weight m0, under the assumptions that m0 is positive on a set of positive Lebesgue measure and $\int_\Omega m_0\,dx \lt 0$, we prove the existence and a characterization of minimizers of $\lambda_1(m)$ and the non-existence of maximizers. Finally, we show that, if Ω is a cylinder, then every minimizer is monotone with respect to the direction of the generatrix. In the context of the population dynamics, this kind of problems arise from the question of determining the optimal spatial location of favourable and unfavourable habitats for a population to survive.

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 obtain Wiman–Valiron type inequalities for random entire functions and for random analytic functions on the unit disk that improve a classical result of Erdős and Rényi and recent results of Kuryliak and Skaskiv. Our results are then applied to linear dynamics: we obtain rates of growth, outside some exceptional set, for analytic functions that are frequently hypercyclic for an arbitrary chaotic weighted backward shift.

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.