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...

We define partial spectral integrals $S_R$ on the Heisenberg group by means of localizations to isotropic or anisotropic dilates of suitable star-shaped subsets V containing the joint spectrum of the partial sub-Laplacians and the central derivative. Under the assumption that an L²-function f lies in the logarithmic Sobolev space given by $log(2+L_{\alpha })f\in L²$, where $L_{\alpha }$ is a suitable “generalized” sub-Laplacian associated to the dilation structure, we show that $S_{R}f\left(x\right)$ converges a.e. to f(x) as R → ∞.

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$\left(2\right)$.

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...

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.

By using the notion of contraction of Lie groups, we transfer $L^p-L²$ estimates for joint spectral projectors from the unit complex sphere $S^{2n+1}$ in ${ℂ}^{n+1}$ to the reduced Heisenberg group hⁿ. In particular, we deduce some estimates recently obtained by H. Koch and F. Ricci on hⁿ. As a consequence, we prove, in the spirit of Sogge’s work, a discrete restriction theorem for the sub-Laplacian L on hⁿ.

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...

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...

Let B be a complex topological unital algebra. The left joint spectrum of a set S ⊂ B is defined by the formula ${\sigma }_l\left(S\right)$ = ${\left(\lambda \left(s\right)\right)}_{s\in S}\in {ℂ}^S|{s-\lambda \left(s\right)}_{s\in S}$ generates a proper left ideal$.$Using the Schur lemma and the Gelfand-Mazur theorem we prove that ${\sigma }_l\left(S\right)$ has the spectral mapping property for sets S of pairwise commuting elements if (i) B is an m-convex algebra with all maximal left ideals closed, or (ii) B is a locally convex Waelbroeck algebra. The right ideal version of this result is also valid.

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...

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...

For a real central arrangement $𝒜$, Salvetti introduced a construction of a finite complex Sal$\left(𝒜\right)$ which is homotopy equivalent to the complement of the complexified arrangement in [Sal87]. For the braid arrangement ${𝒜}_{k-1}$, the Salvetti complex Sal$\left({𝒜}_{k-1}\right)$ serves as a good combinatorial model for the homotopy type of the configuration space $F(ℂ,k)$ of $k$ points in $C$, which is homotopy equivalent to the space $C_2\left(k\right)$ of k little $2$-cubes. Motivated by the importance of little cubes in homotopy theory, especially in...

Let ${\mu }_{n-1}\left(G\right)$ be the algebraic connectivity, and let ${\mu }_1\left(G\right)$ be the Laplacian spectral radius of a $k$-connected graph $G$ with $n$ vertices and $m$ edges. In this paper, we prove that $${\mu }_{n-1}\left(G\right)\ge \frac{2n{k}^2}{(n(n-1)-2m)(n+k-2)+2k^2},$$ with equality if and only if $G$ is the complete graph $K_n$ or $K_n-e$. Moreover, if $G$ is non-regular, then $${\mu }_1\left(G\right)<2\Delta -\frac{2(n\Delta -2m)k^2}{2(n\Delta -2m)(n^2-2n+2k)+n{k}^2},$$ where $\Delta$ stands for the maximum degree of $G$. Remark that in some cases, these two inequalities improve some previously known results.

We give a necessary and a sufficient condition for a subset $S$ of a locally convex Waelbroeck algebra $𝒜$ to have a non-void left joint spectrum ${\sigma }_l\left(S\right).$ In particular, for a Lie subalgebra $L\subset 𝒜$ we have ${\sigma }_l\left(L\right)\ne \varnothing$ if and only if $[L,L]$ generates in $𝒜$ a proper left ideal. We also obtain a version of the spectral mapping formula for a modified left joint spectrum. Analogous theorems for the right joint spectrum and the Harte spectrum are also valid.

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 consider the quantization of inversion in commutative p-normed quasi-Banach algebras with unit. The standard questions considered for such an algebra A with unit e and Gelfand transform x ↦ x̂ are: (i) Is $K_{\nu }=sup{\left|\right|(e-x)}^{-1}{\left|\right|}_p{:x\in A,\left|\right|x\left|\right|}_p\le 1,max\left|x̂\right|\le \nu$ bounded, where ν ∈ (0,1)? (ii) For which δ ∈ (0,1) is $C_{\delta }=sup\left|\right|x^{-1}{\left|\right|}_p{:x\in A,\left|\right|x\left|\right|}_p\le 1,min\left|x̂\right|\ge \delta$ bounded? Both questions are related to a “uniform spectral radius” of the algebra, $r_{\infty }\left(A\right)$, introduced by Björk. Question (i) has an affirmative answer if and only if $r_{\infty }\left(A\right)<1$, and this result is extended to more general nonlinear extremal...