We consider operators acting in the space C(X) (X is a compact topological space) of the form $Au\left(x\right)=\left({\sum }_{k=1}^N{e}^{{\phi }_k}{T}_{{\alpha }_k}\right)u\left(x\right)={\sum }_{k=1}^N{e}^{{\phi }_k\left(x\right)}u\left({\alpha }_k\left(x\right)\right)$, u ∈ C(X), where ${\phi }_k\in C\left(X\right)$ and ${\alpha }_k:X\to X$ are given continuous mappings (1 ≤ k ≤ N). A new formula on the logarithm of the spectral radius r(A) is obtained. The logarithm of r(A) is defined as a nonlinear functional λ depending on the vector of functions $\phi ={\left({\phi }_k\right)}_{k=1}^N$. We prove that $ln\left(r\left(A\right)\right)=\lambda \left(\phi \right)=ma{x}_{\nu \in Mes}{\sum }_{k=1}^N{\int }_X{\phi }_{k}d{\nu }_k-\lambda *\left(\nu \right)$, where Mes is the set of all probability vectors of measures $\nu ={\left({\nu }_k\right)}_{k=1}^N$ on X × 1,..., N and λ* is some convex lower-semicontinuous functional on...

For a set $\{A,B,C,...\}$ of graphs, an $\{A,B,C,...\}$-factor of a graph $G$ is a spanning subgraph $F$ of $G$, where each component of $F$ is contained in $\{A,B,C,...\}$. It is very interesting to investigate the existence of factors in a graph with given minimum degree from the prospective of eigenvalues. We first propose a tight sufficient condition in terms of the $Q$-spectral radius for a graph involving minimum degree to contain a star factor. Moreover, we also present tight sufficient conditions based on the $Q$-spectral radius...

Relations between spectral synthesis in the Fourier algebra A(G) of a compact group G and the concept of operator synthesis due to Arveson have been studied in the literature. For an A(G)-submodule X of VN(G), X-synthesis in A(G) has been introduced by E. Kaniuth and A. Lau and studied recently by the present authors. To any such X we associate a $V^{\infty }\left(G\right)$-submodule X̂ of ℬ(L²(G)) (where $V^{\infty }\left(G\right)$ is the weak-* Haagerup tensor product $L^{\infty }\left(G\right){\otimes }_{w*h}{L}^{\infty }\left(G\right)$), define the concept of X̂-operator synthesis and prove that a...

The work is dedicated to investigating a limiting procedure for extending “local” integral operator equalities to “global” ones in the sense explained below, and to applying it to obtaining generalizations of the Newton-Leibniz formula for operator-valued functions for a wide class of unbounded operators. The integral equalities considered have the form $g\left(R_F\right)\int f_x\left(R_F\right)d\mu \left(x\right)=h\left(R_F\right)$. (1) They involve functions of the kind $X\ni x\mapsto f_x\left(R_F\right)\in B\left(F\right)$, where X is a general locally compact space, F runs over a suitable class of Banach subspaces...

We show that the essential spectral radius ${\varrho }_e\left(T\right)$ of T ∈ B(H) can be calculated by the formula ${\varrho }_e\left(T\right)$ = inf${ℱ}_{♯·♯}\left(XT{X}^{-1}\right)$: X an invertible operator, where ${ℱ}_{♯·♯}\left(T\right)$ is a Φ₁-perturbation function introduced by Mbekhta [J. Operator Theory 51 (2004)]. Also, we show that if ${}_{♯·♯}\left(T\right)$ is a Φ₂-perturbation function [loc. cit.] and if T is a Fredholm operator, then $dist(0,{\sigma }_e\left(T\right))$ = sup${}_{♯·♯}\left(XT{X}^{-1}\right)$: X an invertible operator.

It is known that the nonlinear nonhomogeneous backward Cauchy problem $u_t\left(t\right)+Au\left(t\right)=f(t,u\left(t\right))$, $0\le t<\tau$ with $u\left(\tau \right)=\phi$, where $A$ is a densely defined positive self-adjoint unbounded operator on a Hilbert space, is ill-posed in the sense that small perturbations in the final value can lead to large deviations in the solution. We show, under suitable conditions on $\phi$ and $f$, that a solution of the above problem satisfies an integral equation involving the spectral representation of $A$, which is also ill-posed. Spectral truncation...

Let T be a linear operator on a Banach space X with $supₙ\left|\right|Tⁿ/n^w\left|\right|<\infty$ for some 0 ≤ w < 1. We show that the following conditions are equivalent: (i) $n^{-1}{\sum }_{k=0}^{n-1}{T}^k$ converges uniformly; (ii) $cl(I-T)X=z\in X:li{m}_n{\sum }_{k=1}^n{T}^{k}z/kexists$.

Associated to an Hadamard matrix $H\in M_N\left(ℂ\right)$ is the spectral measure μ ∈ [0,N] of the corresponding Hopf image algebra, A = C(G) with $G\subset S{⁺}_N$. We study a certain family of discrete measures ${\mu }^r\in [0,N]$, coming from the idempotent state theory of G, which converge in Cesàro limit to μ. Our main result is a duality formula of type ${\int }_0^N{(x/N)}^{p}d{\mu }^r\left(x\right)={\int }_0^N{(x/N)}^{r}d{\nu }^p\left(x\right)$, where ${\mu }^r,{\nu }^r$ are the truncations of the spectral measures μ,ν associated to $H,H^t$. We also prove, using these truncations ${\mu }^r,{\nu }^r$, that for any deformed Fourier matrix $H=F_M{\otimes }_Q{F}_N$ we have μ = ν.

A Banach space $X$ has Pełczyński’s property (V) if for every Banach space $Y$ every unconditionally converging operator $T:X\to Y$ is weakly compact. H. Pfitzner proved that $C^{*}$-algebras have Pełczyński’s property (V). In the preprint (Krulišová, (2015)) the author explores possible quantifications of the property (V) and shows that $C\left(K\right)$ spaces for a compact Hausdorff space $K$ enjoy a quantitative version of the property (V). In this paper we generalize this result by quantifying Pfitzner’s theorem. Moreover,...

Let n ≥ 1, d = 2n, and let (x,y) ∈ ℝⁿ × ℝⁿ be a generic point in ℝ²ⁿ. The twisted Laplacian $L=-1/2{\sum }_{j=1}^n[({\partial }_{x_j}+i{y}_j)²+({\partial }_{y_j}-i{x}_j)²]$ has the spectrum n + 2k = λ²: k a nonnegative integer. Let $P_{\lambda }$ be the spectral projection onto the (infinite-dimensional) eigenspace. We find the optimal exponent ϱ(p) in the estimate $\left|\right|P_{\lambda }{u\left|\right|}_{L^p\left({ℝ}^d\right)}\lesssim {\lambda }^{\varrho \left(p\right)}{\left|\right|u\left|\right|}_{L²\left({ℝ}^d\right)}$ for all p ∈ [2,∞], improving previous partial results by Ratnakumar, Rawat and Thangavelu, and by Stempak and Zienkiewicz. The expression for ϱ(p) is ϱ(p) = 1/p -1/2 if 2 ≤ p ≤ 2(d+1)/(d-1), ϱ(p) = (d-2)/2 - d/p...

Given a topological space $X$, let $𝒰X$ and ${\eta }_X:X\to 𝒰X$ denote, respectively, the Salbany compactification of $X$ and the compactification map called the Salbany map of $X$. For every continuous function $f:X\to Y$, there is a continuous function $𝒰f:𝒰X\to 𝒰Y$, called the Salbany lift of $f$, satisfying $\left(𝒰f\right)\circ {\eta }_X={\eta }_Y\circ f$. If a continuous function $f:X\to Y$ has a stably compact codomain $Y$, then there is a Salbany extension $F:𝒰X\to Y$ of $f$, not necessarily unique, such that $F\circ {\eta }_X=f$. In this paper, we give a condition on a space such that its Salbany map is open. In...

We prove that for the spectral radius of a weighted composition operator $a{T}_{\alpha }$, acting in the space $L^p(X,,\mu )$, the following variational principle holds: $lnr\left(a{T}_{\alpha }\right)=ma{x}_{\nu \in M{¹}_{\alpha ,e}}{\int }_{X}ln\left|a\right|d\nu$, where X is a Hausdorff compact space, α: X → X is a continuous mapping preserving a Borel measure μ with suppμ = X, $M{¹}_{\alpha ,e}$ is the set of all α-invariant ergodic probability measures on X, and a: X → ℝ is a continuous and ${}_{\infty }$-measurable function, where ${}_{\infty }={\bigcap }_{n=0}^{\infty }{\alpha }^{-n}\left(\right)$. This considerably extends the range of validity of the above formula, which was previously known...

We establish the spectral gap property for dense subgroups of SU$\left(d\right)$ $(d\ge 2)$, generated by finitely many elements with algebraic entries; this result was announced in [BG3]. The method of proof differs, in several crucial aspects, from that used in [BG] in the case of SU$\left(2\right)$.

Let G be a locally compact group. We use the canonical operator space structure on the spaces $L^p\left(G\right)$ for p ∈ [1,∞] introduced by G. Pisier to define operator space analogues $O{A}_p\left(G\right)$ of the classical Figà-Talamanca-Herz algebras $A_p\left(G\right)$. If p ∈ (1,∞) is arbitrary, then $A_p\left(G\right)\subset O{A}_p\left(G\right)$ and the inclusion is a contraction; if p = 2, then OA₂(G) ≅ A(G) as Banach spaces, but not necessarily as operator spaces. We show that $O{A}_p\left(G\right)$ is a completely contractive Banach algebra for each p ∈ (1,∞), and that $O{A}_q\left(G\right)\subset O{A}_p\left(G\right)$ completely contractively...

The moduli space of rank-$n$ commutative algebras equipped with an ordered basis is an affine scheme ${𝔅}_n$ of finite type over $\mathbb{Z}$, with geometrically connected fibers. It is smooth if and only if $n\le 3$. It is reducible if $n\ge 8$ (and the converse holds, at least if we remove the fibers above $2$ and $3$). The relative dimension of ${𝔅}_n$ is $\frac{2}{27}{n}^3+O\left(n^{8/3}\right)$. The subscheme parameterizing étale algebras is isomorphic to ${GL}_n/S_n$, which is of dimension only $n^2$. For $n\ge 8$, there exist algebras that are not limits of étale algebras. The dimension...

Let $G$ be a graph of order $n$ and $\lambda \left(G\right)$ the spectral radius of its adjacency matrix. We extend some recent results on sufficient conditions for Hamiltonian paths and cycles in $G$. One of the main results of the paper is the following theorem: Let $k\ge 2,$ $n\ge k^3+k+4,$ and let $G$ be a graph of order $n$, with minimum degree $\delta \left(G\right)\ge k.$ If $$\lambda \left(G\right)\ge n-k-1,$$ then $G$ has a Hamiltonian cycle, unless $G=K_1\vee (K_{n-k-1}+K_k)$ or $G=K_k\vee (K_{n-2k}+{\overline{K}}_k).$

In this work we study the problem $u_t{-div\left(\right|x|}^{-2\gamma }\nabla u)=\lambda \frac{u^{\alpha }}{{\left|x\right|}^{2(\gamma +1)}}+f$ in $\Omega \times (0,T)$, $u\ge 0$ in $\Omega \times (0,T)$, $u=0$ on $\partial \Omega \times (0,T)$, $u(x,0)=u_0\left(x\right)$ in $\Omega$, $\Omega \subset {ℝ}^N$ $(N\ge 2)$ is a bounded regular domain such that $0\in \Omega$, $\lambda >0$, $\alpha >0$, $-\infty <\gamma <\left(N-2\right)/2$, $f$ and $u_0$ are positive functions such that $f\in L^1(\Omega \times (0,T))$ and $u_0\in L^1\left(\Omega \right)$. The main points under analysis are: (i) spectral instantaneous and complete blow-up related to the Harnack inequality in the case $\alpha =1$, $1+\gamma >0$; (ii) the nonexistence of solutions if $\alpha >1$, $1+\gamma >0$; (iii) a uniqueness result for weak solutions (in the distribution sense); (iv) further results on existence of weak solutions...

Let K be an algebraically closed field. Let (Q,Sp,I) be a skewed-gentle triple, and let $(Q^{sg},I^{sg})$ and $(Q^g,I^g)$ be the corresponding skewed-gentle pair and the associated gentle pair, respectively. We prove that the skewed-gentle algebra $K{Q}^{sg}/⟨I^{sg}⟩$ is singularity equivalent to KQ/⟨I⟩. Moreover, we use (Q,Sp,I) to describe the singularity category of $K{Q}^g/⟨I^g⟩$. As a corollary, we find that $gldimK{Q}^{sg}/⟨I^{sg}⟩<\infty$ if and only if $gldimKQ/⟨I⟩<\infty$ if and only if $gldimK{Q}^g/⟨I^g⟩<\infty$.

The imbalance of an edge $e=\{u,v\}$ in a graph is defined as $i\left(e\right)=\left|d\right(u)-d(v\left)\right|$, where $d(·)$ is the vertex degree. The irregularity $I\left(G\right)$ of $G$ is then defined as the sum of imbalances over all edges of $G$. This concept was introduced by Albertson who proved that $I\left(G\right)\le 4n^3/27$ (where $n=\left|V\right(G\left)\right|$) and obtained stronger bounds for bipartite and triangle-free graphs. Since then a number of additional bounds were given by various authors. In this paper we prove a new upper bound, which improves a bound found by Zhou and Luo in 2008. Our bound involves...