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...

Let $A$ be a bounded linear operator in a complex separable Hilbert space $\mathscr{H}$, and $S$ be a selfadjoint operator in $\mathscr{H}$. Assuming that $A-S$ belongs to the Schatten-von Neumann ideal ${𝒮}_p$ $(p>1),$ we derive a bound for ${\sum }_k{|\mathrm{R}{\lambda }_k\left(A\right)-{\lambda }_k\left(S\right)|}^p$, where ${\lambda }_k\left(A\right)$ $(k=1,2,\cdots )$ are the eigenvalues of $A$. Our results are formulated in terms of the “extended” eigenvalue sets in the sense introduced by T. Kato. In addition, in the case $p=2$ we refine the Weyl inequality between the real parts of the eigenvalues of $A$ and the eigenvalues...

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...

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 $p,q\in ℝ$ with $p-q\ge 0$, $\sigma =\frac{1}{2}(p+q-1)$ and $s=\frac{1}{2}(1-p+q)$, and let $${𝒟}_m(x;p,q)={𝒟}_0(x;p,q)+\sum _{k=1}^m\frac{B_{2k}\left(s\right)}{2k{(x+\sigma )}^{2k}},$$ where $${𝒟}_0(x;p,q)=\frac{\psi (x+p)+\psi (x+q)}{2}-ln(x+\sigma ).$$ We establish the asymptotic expansion $${𝒟}_0(x;p,q)\sim -\sum _{n=1}^{\infty }\frac{B_{2n}\left(s\right)}{2n{(x+\sigma )}^{2n}}\text{as}x\to \infty ,$$ where $B_{2n}\left(s\right)$ stands for the Bernoulli polynomials. Further, we prove that the functions ${(-1)}^m{𝒟}_m(x;p,q)$ and ${(-1)}^{m+1}{𝒟}_m(x;p,q)$ are completely monotonic in $x$ on $(-\sigma ,\infty )$ for every $m\in {\mathbb{N}}_0$ if and only if $p-q\in [0,\frac{1}{2}]$ and $p-q=1$, respectively. This not only unifies the two known results but also yields some new results.

We construct some nondecreasing quantities associated to the first eigenvalue of $-{\Delta }_{\phi }+cR$ $(c\ge \frac{1}{2}(n-2)/(n-1))$ along the Yamabe flow, where ${\Delta }_{\phi }$ is the Witten-Laplacian operator with a $C^2$ function $\phi$. We also prove a monotonic result on the first eigenvalue of $-{\Delta }_{\phi }+\frac{1}{4}(n/(n-1))R$ along the Yamabe flow. Moreover, we establish some nondecreasing quantities for the first eigenvalue of $-{\Delta }_{\phi }+c{R}^a$ with $a\in (0,1)$ along the Yamabe flow.

Let $a=(a_1,a_2,...,a_n)$ be a nonincreasing sequence of positive real numbers. Denote by $S=\{1,2,...,n\}$ the index set and by $J_k=\{I=\{r_1,r_2,...,r_k\}$, $1\le r_1<r_2<\cdots <r_k\le n\}$ the set of all subsets of $S$ of cardinality $k$, $1\le k\le n-1$. In addition, denote by $a_I=a_{r_1}+a_{r_2}+\cdots +a_{r_k}$, $1\le k\le n-1$, $1\le r_1<r_2<\cdots <r_k\le n$, the sum of $k$ arbitrary elements of sequence $a$, where $a_{I_1}=a_1+a_2+\cdots +a_k$ and $a_{I_n}=a_{n-k+1}+a_{n-k+2}+\cdots +a_n$. We consider bounds of the quantities $R{S}_k\left(a\right)=a_{I_1}/a_{I_n}$, $L{S}_k\left(a\right)=a_{I_1}-a_{I_n}$ and $S_{k,\alpha }\left(a\right)={\sum }_{I\in J_k}{a}_I^{\alpha }$ in terms of $A={\sum }_{i=1}^n{a}_i$ and $B={\sum }_{i=1}^n{a}_i^2$. Then we use the obtained results to generalize some results regarding Laplacian and normalized Laplacian eigenvalues of graphs.

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).

Motivated by the fundamental theorem of calculus, and based on the works of W. Feller as well as M. Kac and M. G. Kreĭn, given an atomless Borel probability measure $\eta$ supported on a compact subset of $ℝ$ U. Freiberg and M. Zähle introduced a measure-geometric approach to define a first order differential operator ${\nabla }_{\eta }$ and a second order differential operator ${\Delta }_{\eta }$, with respect to $\eta$. We generalize this approach to measures of the form $\eta :=\nu +\delta$, where $\nu$ is non-atomic and $\delta$ is finitely supported. We determine...

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 $\lambda >0$.