Harper’s operator is defined on ${\ell }^2\left(Z\right)$ by $$H_{\theta }\xi \left(n\right)=\xi (n+1)+\xi (n-1)+2cosn\theta \xi \left(n\right),$$ where $\theta \in [0,\pi ]$. We show that the norm of $\parallel H_{\theta }\parallel$ is less than or equal to $2\sqrt{2}$ for $\pi /2\le \theta \le \pi$. This solves a conjecture stated in [1]. A general formula for estimating the norm of self-adjoint tridiagonal infinite matrices is also derived.

Let $ℬ=f:ℝ\to ℝ:fisbounded$, and $ℳ=f:ℝ\to ℝ:fisLebesguemeasurable$. We show that there is a linear operator $\Phi :ℬ\to ℳ$ such that Φ(f)=f a.e. for every $f\in ℬ\cap ℳ$, and Φ commutes with all translations. On the other hand, if $\Phi :ℬ\to ℳ$ is a linear operator such that Φ(f)=f for every $f\in ℬ\cap ℳ$, then the group $G_{\Phi }$ = a ∈ ℝ:Φ commutes with the translation by a is of measure zero and, assuming Martin’s axiom, is of cardinality less than continuum. Let Φ be a linear operator from ${ℂ}^{ℝ}$ into the space of complex-valued measurable functions. We show that if Φ(f) is non-zero for every $f\left(x\right)=e^{cx}$, then $G_{\Phi }$ must...

We prove that the absolutely continuous part of the periodic Jacobi operator does not change (modulo unitary equivalence) under additive perturbations by compact Jacobi operators with weights and diagonals defined in terms of the Stolz classes of slowly oscillating sequences. This result substantially generalizes many previous results, e.g., the one which can be obtained directly by the abstract trace class perturbation theorem of Kato-Rosenblum. It also generalizes several results concerning perturbations...

We give examples of polynomials p(n) orthonormal with respect to a measure μ on ⨍ such that the sequence {p(n,x)} has exponential lower bound for some points x of supp μ. Moreover, the set of such points is dense in the support of μ.

For the difference equation $\left(ϵ\right)x_{n+1}=A{x}_n+ϵ{\sum }_{k=-\infty }^n{R}_{n-k}{x}_k$,where $x_n\in Y,Y$  is a Banach space, $ϵ$ is a parameter and  $A$  is a linear, bounded operator. A sufficient condition for the existence of a unique special solution  $y={\left\{y_n\right\}}_{n=-\infty }^{\infty }$  passing through the point  $x_0\in Y$  is proved. This special solution converges to the solution of the equation (0) as  $ϵ\to 0$.

This paper initially considers a class of unbounded Jacobi matrices defined by an increasing sequence of repeated weights. Spectral parameters are then introduced in various ways to allow the authors to study the nature and location of the spectrum as a function of these parameters.

Let X be an arbitrary Abelian group and E a Banach space. We consider the difference-operators ∆n defined by induction:(∆f)(x;y) = f(x+y) - f(x), (∆nf)(x;y1,...,yn) = (∆n-1(∆f)(.;y1)) (x;y2,...,yn)(n = 2,3,4,..., ∆1=∆, x,yi belonging to X, i = 1,2,...,n; f: X --> E).Considering the difference equation (∆nf)(x;y1,y2,...,yn) = d(x;y1,y2,...,yn) with independent variable increments, the most general solution is given explicitly if d: X x Xn --> E is a given bounded function. Also the...

The Embry conditions are a set of positivity conditions that characterize subnormal operators (on Hilbert spaces) whose theory is closely related to the theory of positive definite functions on the additive semigroup ℕ of non-negative integers. Completely hyperexpansive operators are the negative definite counterpart of subnormal operators. We show that completely hyperexpansive operators are characterized by a set of negativity conditions, which are the natural analog of the Embry conditions for...