This note proves that the annihilation operator of a quantum harmonic oscillator admits an invariant distributionally $\epsilon$-scrambled linear manifold for any . This is a positive answer to Question 1 by Wu and Chen (2013).

Grosse-Erdmann and Kim recently introduced the notion of bihypercyclicity for studying the existence of dense orbits under bilinear operators. We propose an alternative notion of orbit for $N$-linear operators that is inspired by difference equations. Under this new notion, every separable infinite dimensional Fréchet space supports supercyclic $N$-linear operators, for each $N\ge 2$. Indeed, the nonnormable spaces of entire functions and the countable product of lines support $N$-linear operators with residual sets of hypercyclic vectors, for $N=2$.

For two complex Banach spaces $X$ and $Y$, in this paper, we study the generalized spectrum ${ℳ}_b(X,Y)$ of all nonzero algebra homomorphisms from ${\mathscr{H}}_b(X)$, the algebra of all bounded type entire functions on $X$, into ${\mathscr{H}}_b(Y)$. We endow ${ℳ}_b(X,Y)$ with a structure of Riemann domain over $ℒ(X^{\mathrm{*}},Y^{\mathrm{*}})$ whenever $X$ is symmetrically regular. The size of the fibers is also studied. Following the philosophy of (Aron et al., 1991), this is a step to study the set ${ℳ}_{b,\mathrm{\infty }}(X,B_Y)$ of all nonzero algebra homomorphisms from ${\mathscr{H}}_b(X)$ into ${\mathscr{H}}_{\mathrm{\infty }}(B_Y)$ of bounded holomorphic functions on the open unit ball of $Y$ and ${ℳ}_{\mathrm{\infty }}(B_X,B_Y)$ of all nonzero algebra homomorphisms from ${\mathscr{H}}_{\mathrm{\infty }}(B_X)$ into ${\mathscr{H}}_{\mathrm{\infty }}(B_Y)$.

A bounded operator $T$ on a Banach space $X$ is convex cyclic if there exists a vector $x$ such that the convex hull generated by the orbit ${\left\{T^{n}x\right\}}_{n\ge 0}$ is dense in $X$. In this note we study some questions concerned with convex-cyclic operators. We provide an example of a convex-cyclic operator $T$ such that the power $T^n$ fails to be convex cyclic. Using this result we solve three questions posed by Rezaei (2013).

We study norm attaining properties of the Arens extensions of multilinear forms defined on Banach spaces. Among other related results, we construct a multilinear form on ${\ell }_1$ with the property that only some fixed Arens extensions determined a priori attain their norms. We also study when multilinear forms can be approximated by ones with the property that only some of their Arens extensions attain their norms.

We study the Bishop-Phelps-Bollobás property for numerical radius (in short, BPBp-nu) and find sufficient conditions for Banach spaces to ensure the BPBp-nu. Among other results, we show that $L_1\left(\mu \right)$-spaces have this property for every measure μ. On the other hand, we show that every infinite-dimensional separable Banach space can be renormed to fail the BPBp-nu. In particular, this shows that the Radon-Nikodým property (even reflexivity) is not enough to get BPBp-nu.

We study the asymptotic behaviors of the discrete eigenvalue of Schrödinger operator $P(\lambda )=P_0+\lambda V$ with $P_0=-\mathrm{\Delta }+q\left(\theta \right)/r^2.$ We obtain the leading terms of discrete eigenvalues of $P(\lambda )$ when the eigenvalues tend to 0. In particular, we obtain the asymptotic behaviors of eigenvalues when ${(P_0-\alpha )}^{-1}$ has singularity at $\alpha =0$.

Localization operators in the discrete setting are used to obtain information on a signal f from the knowledge on the support of its short time Fourier transform. In particular, the extremal functions of the uncertainty principle for the discrete short time Fourier transform are characterized and their connection with functions that generate a time-frequency basis is studied.

Well-known Banach space results (e.g., due to J. Koliha and Y. Katznelson/L. Tzafriri), which relate conditions on the spectrum of a bounded operator $T$ to the operator norm convergence of certain sequences of operators generated by $T$, are extended to the class of quojection Fréchet spaces. These results are then applied to establish various mean ergodic theorems for continuous operators acting in such Fréchet spaces and which belong to certain operator ideals, for example, compact, weakly compact, and Montel.