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We show that for self-adjoint Jacobi matrices and Schrödinger operators, perturbed by dissipative potentials in 
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\ell ^1({\mathbb{N}})$ and 
$L^1(0,\infty )$ respectively, the finite section method does not omit any points of the spectrum. In the Schrödinger case two different approaches are presented. Many aspects of the proofs can be expected to carry over to higher dimensions, particularly for absolutely continuous spectrum.
Let 
$\alpha : 1, 1, \sqrt{x} , \mathop{( \sqrt{u} , \sqrt{v} , \sqrt{w} )}\nolimits ^{\wedge } $ be a backward 3-step extension of a recursively generated weighted sequence of positive real numbers with 
$1\leq x\leq u\leq v\leq w$ and let 
${W}_{\alpha } $ be the associated weighted shift with weight sequence 
$\alpha $. The set of positive real numbers 
$x$ such that 
${W}_{\alpha } $ is quadratically hyponormal for some 
$u, v$ and 
$w$ is described, solving an open problem due to Curto and Jung [‘Quadratically hyponormal weighted shifts with two equal weights’, Integr. Equ. Oper. Theory 37 (2000), 208–231].
Given two arbitrary sequences 
$({\lambda }_{j} )_{j\geq 1} $ and 
$({\mu }_{j} )_{j\geq 1} $ of real numbers satisfying we prove that there exists a unique sequence 
$$\begin{eqnarray*}\displaystyle \vert {\lambda }_{1} \vert \gt \vert {\mu }_{1} \vert \gt \vert {\lambda }_{2} \vert \gt \vert {\mu }_{2} \vert \gt \cdots \gt \vert {\lambda }_{j} \vert \gt \vert {\mu }_{j} \vert \rightarrow 0, &&\displaystyle\end{eqnarray*}$$
$c= ({c}_{n} )_{n\in { \mathbb{Z} }_{+ } } $, real valued, such that the Hankel operators 
${\Gamma }_{c} $ and 
${\Gamma }_{\tilde {c} } $ of symbols 
$c= ({c}_{n} )_{n\geq 0} $ and 
$\tilde {c} = ({c}_{n+ 1} )_{n\geq 0} $, respectively, are selfadjoint compact operators on 
${\ell }^{2} ({ \mathbb{Z} }_{+ } )$ and have the sequences 
$({\lambda }_{j} )_{j\geq 1} $ and 
$({\mu }_{j} )_{j\geq 1} $, respectively, as non-zero eigenvalues. Moreover, we give an
explicit formula for 
$c$ and we describe the kernel of 
${\Gamma }_{c} $ and of 
${\Gamma }_{\tilde {c} } $ in terms of the sequences 
$({\lambda }_{j} )_{j\geq 1} $ and 
$({\mu }_{j} )_{j\geq 1} $. More generally, given two arbitrary sequences 
$({\rho }_{j} )_{j\geq 1} $ and 
$({\sigma }_{j} )_{j\geq 1} $ of positive numbers satisfying we describe the set of sequences 
$$\begin{eqnarray*}\displaystyle {\rho }_{1} \gt {\sigma }_{1} \gt {\rho }_{2} \gt {\sigma }_{2} \gt \cdots \gt {\rho }_{j} \gt {\sigma }_{j} \rightarrow 0, &&\displaystyle\end{eqnarray*}$$
$c= ({c}_{n} )_{n\in { \mathbb{Z} }_{+ } } $ of complex numbers such that the Hankel operators 
${\Gamma }_{c} $ and 
${\Gamma }_{\tilde {c} } $ are compact on 
${\ell }^{2} ({ \mathbb{Z} }_{+ } )$ and have sequences 
$({\rho }_{j} )_{j\geq 1} $ and 
$({\sigma }_{j} )_{j\geq 1} $, respectively, as non-zero singular values.
Let A be a unital C*-algebra with the canonical (H) C*-bundle 
over the maximal ideal space of the centre of A, and let E(A) be the set of all elementary operators on A. We consider derivations on A which lie in the completely bounded norm closure of E(A), and show that such derivations are necessarily inner in the case when each fibre of is a prime C*-algebra. We also consider separable C*-algebras A for which is an (F) bundle. For these C*-algebras we show that the following conditions are equivalent: E(A) is closed in the operator norm; A as a Banach module over its centre is topologically finitely generated; fibres of have uniformly finite dimensions, and each restriction bundle of over a set where its fibres are of constant dimension is of finite type as a vector bundle.
Let 
$\phi $ and 
$\psi $ be analytic maps on the open unit disk 
$D$ such that 
$\phi (D) \subset D$. Such maps induce a weighted composition operator 
$C_{\phi ,\psi }$ acting on weighted Banach spaces of type 
$H^{\infty }$or on weighted Bergman spaces, respectively. We study when such operators are order bounded.
We study the boundedness and compactness of Toeplitz operators Ta on Bergman spaces 
, 1 < p < ∞. The novelty is that we allow distributional symbols. It turns out that the belonging of the symbol to a weighted Sobolev space 
of negative order is sufficient for the boundedness of Ta. We show the natural relation of the hyperbolic geometry of the disc and the order of the distribution. A corresponding sufficient condition for the compactness is also derived.
We use induction and interpolation techniques to prove that a composition operator induced by a map ϕ is bounded on the weighted Bergman space 
of the right half-plane if and only if ϕ fixes the point at ∞ non-tangentially and if it has a finite angular derivative λ there. We further prove that in this case the norm, the essential norm and the spectral radius of the operator are all equal and are given by λ(2+α)/2.
