Let $B\left(X\right)$ be the algebra of all bounded linear operators in a complex Banach space $X$. We consider operators $T_1,T_2\in B\left(X\right)$ satisfying the relation ${\sigma }_{T_1}\left(x\right)={\sigma }_{T_2}\left(x\right)$ for any vector $x\in X$, where ${\sigma }_T\left(x\right)$ denotes the local spectrum of $T\in B\left(X\right)$ at the point $x\in X$. We say then that $T_1$ and $T_2$ have the same local spectra. We prove that then, under some conditions, $T_1-T_2$ is a quasinilpotent operator, that is $\parallel {(T_1-T_2)}^n{\parallel }^{1/n}\to 0$ as $n\to \infty$. Without these conditions, we describe the operators with the same local spectra only in some particular cases.

We show a polynomially boundend operator T is similar to a unitary operator if there is a singular unitary operator W and an injection X such that XT = WX. If, in addition, T is of class $C_{\varrho }$, then T itself is unitary.

A two-sided sequence ${\left(cₙ\right)}_{n\in \mathbb{Z}}$ with values in a complex unital Banach algebra is a cosine sequence if it satisfies $c_{n+m}+c_{n-m}=2cₙcₘ$ for any n,m ∈ ℤ with c₀ equal to the unity of the algebra. A cosine sequence ${\left(cₙ\right)}_{n\in \mathbb{Z}}$ is bounded if $su{p}_{n\in \mathbb{Z}}\left|\right|cₙ\left|\right|<\infty$. A (bounded) group decomposition for a cosine sequence $c={\left(cₙ\right)}_{n\in \mathbb{Z}}$ is a representation of c as $cₙ=(bⁿ+b^{-n})/2$ for every n ∈ ℤ, where b is an invertible element of the algebra (satisfying $su{p}_{n\in \mathbb{Z}}\left|\right|bⁿ\left|\right|<\infty$, respectively). It is known that every bounded cosine sequence possesses a universally defined group decomposition, the so-called...

Using interpolation techniques we prove an optimal regularity theorem for the convolution $u\left(t\right)={\int }_0^{t}T\left(t-s\right)f\left(s\right)ds$, where $T\left(t\right)$ is a strongly continuous semigroup in general Banach space. In the case of abstract parabolic problems – that is, when $T\left(t\right)$ is an analytic semigroup – it lets us recover in a unified way previous regularity results. It may be applied also to some non analytic semigroups, such as the realization of the Ornstein-Uhlenbeck semigroup in $L^p\left({\mathbb{R}}^n\right)$, $1<p<\mathrm{\infty }$, in which case it yields new optimal regularity results in fractional...

It is known that there is a continuous linear functional on L ∞ which is not narrow. On the other hand, every order-to-norm continuous AM-compact operator from L ∞(μ) to a Banach space is narrow. We study order-to-norm continuous operators acting from L ∞(μ) with a finite atomless measure μ to a Banach space. One of our main results asserts that every order-to-norm continuous operator from L ∞(μ) to c 0(Γ) is narrow while not every such an operator is AM-compact.

Sharp $L^p$ estimates are proven for oscillatory integrals with phase functions Φ(x,y), (x,y) ∈ X × Y, under the assumption that the canonical relation $C_{\Phi }$ projects to T*X and T*Y with fold singularities.