Let ϕ: [0,1] → [0,1] be a nondecreasing continuous function such that ϕ(x) > x for all x ∈ (0,1). Let the operator $V_{\varphi }:f\left(x\right)\mapsto {\int }_0^{\varphi \left(x\right)}f\left(t\right)dt$ be defined on L₂[0,1]. We prove that $V_{\varphi }$ has a finite number of nonzero eigenvalues if and only if ϕ(0) > 0 and ϕ(1-ε) = 1 for some 0 < ε < 1. Also, we show that the spectral trace of the operator $V_{\varphi }$ always equals 1.

We investigate the behaviour of a sequence ${\lambda }_s$, s = 1,2,..., of eigenvalues of the Dirichlet problem for the p-Laplacian in the domains ${\Omega }_s$, s = 1,2,..., obtained by removing from a given domain Ω a set $E_s$ whose diameter vanishes when s → ∞. We estimate the deviation of ${\lambda }_s$ from the eigenvalue of the limit problem. For the derivation of our results we construct an appropriate asymptotic expansion for the sequence of solutions of the original eigenvalue problem.

A substitution φ is strong Pisot if its abelianization matrix is nonsingular and all eigenvalues except the Perron-Frobenius eigenvalue have modulus less than one. For strong Pisot φ that satisfies a no cycle condition and for which the translation flow on the tiling space ${}_{\phi }$ has pure discrete spectrum, we describe the collection ${}_{\phi }^P$ of pairs of proximal tilings in ${}_{\phi }$ in a natural way as a substitution tiling space. We show that if ψ is another such substitution, then ${}_{\phi }$ and ${}_{\psi }$ are homeomorphic...

We discuss the discrete $p$-Laplacian eigenvalue problem, $$\left\{\begin{array}{c}\Delta \left({\phi }_p\left(\Delta u(k-1)\right)\right)+\lambda a\left(k\right)g\left(u\left(k\right)\right)=0,k\in \{1,2,...,T\},\hfill \\ u\left(0\right)=u(T+1)=0,\hfill \end{array}\right.$$ where $T>1$ is a given positive integer and ${\phi }_p\left(x\right):={\left|x\right|}^{p-2}x$, $p>1$. First, the existence of an unbounded continuum $𝒞$ of positive solutions emanating from $(\lambda ,u)=(0,0)$ is shown under suitable conditions on the nonlinearity. Then, under an additional condition, it is shown that the positive solution is unique for any $\lambda >0$ and all solutions are ordered. Thus the continuum $𝒞$ is a monotone continuous curve globally defined for all $\lambda >0$.

An infinite set A in a space X converges to a point p (denoted by A → p) if for every neighbourhood U of p we have |A∖U| < |A|. We call cS(p,X) = |A|: A ⊂ X and A → p the convergence spectrum of p in X and cS(X) = ⋃cS(x,X): x ∈ X the convergence spectrum of X. The character spectrum of a point p ∈ X is χS(p,X) = χ(p,Y): p is non-isolated in Y ⊂ X, and χS(X) = ⋃χS(x,X): x ∈ X is the character spectrum of X. If κ ∈ χS(p,X) for a compactum X then κ,cf(κ) ⊂ cS(p,X). A selection of our...

We survey recent results concerning estimates of the principal eigenvalue of the Dirichlet $p$-Laplacian and the Navier $p$-biharmonic operator on a ball of radius $R$ in ${ℝ}^N$ and its asymptotics for $p$ approaching $1$ and $\infty$. Let $p$ tend to $\infty$. There is a critical radius $R_C$ of the ball such that the principal eigenvalue goes to $\infty$ for $0<R\le R_C$ and to $0$ for $R>R_C$. The critical radius is $R_C=1$ for any $N\in \mathbb{N}$ for the $p$-Laplacian and $R_C=\sqrt{2N}$ in the case of the $p$-biharmonic operator. When $p$ approaches $1$, the principal eigenvalue...

An operator T acting on a Banach space X has property (gw) if ${\sigma }_a\left(T\right)\setminus {\sigma }_{SBF₊¯}\left(T\right)=E\left(T\right)$, where ${\sigma }_a\left(T\right)$ is the approximate point spectrum of T, ${\sigma }_{SBF₊¯}\left(T\right)$ is the upper semi-B-Weyl spectrum of T and E(T) is the set of all isolated eigenvalues of T. We introduce and study two new spectral properties (v) and (gv) in connection with Weyl type theorems. Among other results, we show that T satisfies (gv) if and only if T satisfies (gw) and $\sigma \left(T\right)={\sigma }_a\left(T\right)$.

Let $(M,g)$ be a compact Riemannian manifold and $P_g$ an elliptic, formally self-adjoint, conformally covariant operator of order $m$ acting on smooth sections of a bundle over $M$. We prove that if $P_g$ has no rigid eigenspaces (see Definition 2.2), the set of functions $f\in C^{\infty }(M,ℝ)$ for which $P_{e^{f}g}$ has only simple non-zero eigenvalues is a residual set in $C^{\infty }(M,ℝ)$. As a consequence we prove that if $P_g$ has no rigid eigenspaces for a dense set of metrics, then all non-zero eigenvalues are simple for a residual set of metrics...

For a linear operator T in a Banach space let ${\sigma }_p\left(T\right)$ denote the point spectrum of T, let ${\sigma }_{p,n}\left(T\right)$ for finite n > 0 be the set of all $\lambda \in {\sigma }_p\left(T\right)$ such that dim ker(T - λ) = n and let ${\sigma }_{p,\infty }\left(T\right)$ be the set of all $\lambda \in {\sigma }_p\left(T\right)$ for which ker(T - λ) is infinite-dimensional. It is shown that ${\sigma }_p\left(T\right)$ is ${ℱ}_{\sigma }$, ${\sigma }_{p,\infty }\left(T\right)$ is ${ℱ}_{\sigma \delta }$ and for each finite n the set ${\sigma }_{p,n}\left(T\right)$ is the intersection of an ${ℱ}_{\sigma }$ set and a ${}_{\delta }$ set provided T is closable and the domain of T is separable and weakly σ-compact. For closed densely defined operators in a separable Hilbert space a more...

We consider simultaneous solutions of operator Sylvester equations $A_{i}X-X{B}_i=C_i$ (1 ≤ i ≤ k), where $(A₁,...,A_k)$ and $(B₁,...,B_k)$ are commuting k-tuples of bounded linear operators on Banach spaces and ℱ, respectively, and $(C₁,...,C_k)$ is a (compatible) k-tuple of bounded linear operators from ℱ to , and prove that if the joint Taylor spectra of $(A₁,...,A_k)$ and $(B₁,...,B_k)$ do not intersect, then this system of Sylvester equations has a unique simultaneous solution.

We show that the spectrum of $\Delta {²}_{p}u+{2\beta ·\nabla \left(\right|\Delta u|}^{p-2}{\Delta u)+|\beta \left|²\right|\Delta u|}^{p-2}\Delta u=\alpha m{\left|u\right|}^{p-2}u$, where $\beta \in {ℝ}^N$, under Navier boundary conditions, contains at least one sequence of eigensurfaces.

We analyse the spectral phase diagram of Schrödinger operators $T+\lambda V$ on regular tree graphs, with $T$ the graph adjacency operator and $V$ a random potential given by $iid$ random variables. The main result is a criterion for the emergence of absolutely continuous $\left(ac\right)$ spectrum due to fluctuation-enabled resonances between distant sites. Using it we prove that for unbounded random potentials $ac$ spectrum appears at arbitrarily weak disorder $(\lambda \ll 1)$ in an energy regime which extends beyond the spectrum of$\sim T$....

Let $\left\{X_{ij}\right\}$, $i,j=\cdots$, be a double array of independent and identically distributed (i.i.d.) real random variables with $E{X}_{11}=\mu$, $E|X_{11}{-\mu |}^2=1$ and $E|X_{11}{|}^{4}lt;\infty$. Consider sample covariance matrices (with/without empirical centering) $𝒮=\frac{1}{n}{\sum }_{j=1}^n({𝐬}_j-\overline{𝐬}){({𝐬}_j-\overline{𝐬})}^T$ and $𝐒=\frac{1}{n}{\sum }_{j=1}^n{𝐬}_j{𝐬}_j^T$, where $\overline{𝐬}=\frac{1}{n}{\sum }_{j=1}^n{𝐬}_j$ and ${𝐬}_j={𝐓}_n^{1/2}{(X_{1j},...,X_{pj})}^T$ with ${\left({𝐓}_n^{1/2}\right)}^2={𝐓}_n$, non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of $𝒮$ and $𝐒$ are different as $n\to \infty$ with $p/n$ approaching a positive constant. Moreover, it is also proved that such a different behavior is not observed in the...

Let $G$ be an undirected connected graph with $n$, $n\ge 3$, vertices and $m$ edges with Laplacian eigenvalues ${\mu }_1\ge {\mu }_2\ge \cdots \ge {\mu }_{n-1}>{\mu }_n=0$. Denote by ${\mu }_I={\mu }_{r_1}+{\mu }_{r_2}+\cdots +{\mu }_{r_k}$, $1\le k\le n-2$, $1\le r_1<r_2<\cdots <r_k\le n-1$, the sum of $k$ arbitrary Laplacian eigenvalues, with ${\mu }_{I_1}={\mu }_1+{\mu }_2+\cdots +{\mu }_k$ and ${\mu }_{I_n}={\mu }_{n-k}+\cdots +{\mu }_{n-1}$. Lower bounds of graph invariants ${\mu }_{I_1}-{\mu }_{I_n}$ and ${\mu }_{I_1}/{\mu }_{I_n}$ are obtained. Some known inequalities follow as a special case.

First, some classic properties of a weighted Frobenius-Perron operator ${𝒫}_{\varphi }^u$ on $L^1\left(\Sigma \right)$ as a predual of weighted Koopman operator $W=u{U}_{\varphi }$ on $L^{\infty }\left(\Sigma \right)$ will be investigated using the language of the conditional expectation operator. Also, we determine the spectrum of ${𝒫}_{\varphi }^u$ under certain conditions.

Consider the $n\times n$ matrix with $(i,j)$’th entry $gcd(i,j)$. Its largest eigenvalue ${\lambda }_n$ and sum of entries $s_n$ satisfy ${\lambda }_n>s_n/n$. Because $s_n$ cannot be expressed algebraically as a function of $n$, we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S. Hong, R. Loewy (2004). We also conjecture that ${\lambda }_n>6{\pi }^{-2}nlogn$ for all $n$. If $n$ is large enough, this follows from F. Balatoni (1969).