On powers of operators with spectrum in cantor sets and spectral synthesis

For $\xi \in \big( 0, \frac{1}{2} \big)$, let $E_{\xi}$ be the perfect symmetric set associated with $\xi$, that is $$E_{\xi} = \Big\{ \exp \Big( 2i \pi (1-\xi) \sum_{n = 1}^{+\infty} \epsilon_{n} \xi^{n-1} \Big) : \, \epsilon_{n} = 0 \textrm{ or } 1 \quad (n \geq 1) \Big\}$$ and $$b(\xi) = \frac{\log{\frac{1}{\xi}} - \log{2}}{2\log{\frac{1}{\xi}} - \log{2}}.$$ Let $q\geq 3$ be an integer and $s$ be a nonnegative real number. We show that any invertible operator $T$ on a Banach space with spectrum contained in $E_{1/q}$ that satisfies \begin{eqnarray*} & & \big\| T^{n} \big\| = O \big( n^{s} \big), \,n \rightarrow +\infty \\ & \textrm{and} & \big\| T^{-n} \big\| = O \big( e^{n^{\beta}} \big), \, n \rightarrow +\infty \textrm{ for some } \beta < b(1/q),\end{eqnarray*} also satisfies the stronger property $\big\| T^{-n} \big\| = O \big( n^{s} \big), \, n \rightarrow +\infty.$ We also show that this result is false for $E_\xi$ when $1/\xi$ is not a Pisot number and that the constant $b(1/q)$ is sharp. As a consequence we prove that, if $\omega$ is a submulticative weight such that $\omega(n)=(1+n)^s, \, (n \geq 0)$ and $C^{-1} (1+|n|)^s \leq \omega(-n) \leq C e^{n^{\beta}},\, (n\geq 0)$, for some constants $C>0$ and $\beta < b( 1/q),$ then $E_{1/q}$ satisfies spectral synthesis in the Beurling algebra of all continuous functions $f$ on the unit circle $\mathbb{T}$ such that $\sum_{n = -\infty}^{+\infty} | \widehat{f}(n) | \omega (n) < +\infty$

Weighted Big Lipschitz algebras of analytic functions and closed ideals

We give the smallest closed ideal with given hull and inner factor for some weighted big Lipschitz algebras of analytic functions

A simple proof for visibility paths in simple polygons

The purpose of this note is to give a simple proof for a necessary and sufficient condition for visibility paths in simple polygons. A visibility path is a curve such that every point inside a simple polygon is visible from at least one point on the path. This result is essential for finding the shortest watchman route inside a simple polygon specially when the route is restricted to curved paths

Query-points visibility constraint minimum link paths in simple polygons

We study the query version of constrained minimum link paths between two points inside a simple polygon $P$ with $n$ vertices such that there is at least one point on the path, visible from a query point. The method is based on partitioning $P$ into a number of faces of equal link distance from a point, called a link-based shortest path map (SPM). Initially, we solve this problem for two given points $s$, $t$ and a query point $q$. Then, the proposed solution is extended to a general case for three arbitrary query points $s$, $t$ and $q$. In the former, we propose an algorithm with $O(n)$ preprocessing time. Extending this approach for the latter case, we develop an algorithm with $O(n^3)$ preprocessing time. The link distance of a $q$-$visible$ path between $s$, $t$ as well as the path are provided in time $O(\log n)$ and $O(m+\log n)$, respectively, for the above two cases, where $m$ is the number of links

Unitary equivalence to truncated Toeplitz operators

In this paper we investigate operators unitarily equivalent to truncated Toeplitz operators. We show that this class contains certain sums of tensor products of truncated Toeplitz operators. In particular, it contains arbitrary inflations of truncated Toeplitz operators; this answers a question posed by Cima, Garcia, Ross, and Wogen

Effects Of Material Parameters On The Stresses In High Temperature Weldments

An integral equation is presented that relates time to equivalent stresses at the interface of two regions in a weldment operating within the creep regime. Taking the creep constitutive equation as the Norton power law, the paper investigates the effects of the stress index and the stress coefficient on the equivalent stresses at the critical interfaces in the weldment

The commutant of an operator with bounded conjugation orbits and C₀‒contractions

Let A be an invertible bounded linear operator on a complex Banach space, {A}′ the commutant of A and Bᴀ the set of all operators T such that 〖sup〗_(ₙ≥₀)∥AⁿTA⁻ⁿ ∥ < +∞. Equality {A}′ = Bᴀ was studied by many authors for differents classes of operators. In this paper we investigate a local version of this equality and the case where A is a C₀–contraction.peerReviewe