Trotter-Kato product formulae in Dixmier ideal

It is shown that for a certain class of the Kato functions the Trotter-Kato product formulae converge in Dixmier ideal C 1,$\infty$ in topology, which is defined by the $\times$ 1,$\infty$-norm. Moreover, the rate of convergence in this topology inherits the error-bound estimate for the corresponding operator-norm convergence. 1 since [24], [14]. Note that a subtle point of this program is the question about the rate of convergence in the corresponding topology. Since the limit of the Trotter-Kato product formula is a strongly continuous semigroup, for the von Neumann-Schatten ideals this topology is the trace-norm $\times$ 1 on the trace-class ideal C 1 (H). In this case the limit is a Gibbs semigroup [25]. For self-adjoint Gibbs semigroups the rate of convergence was estimated for the first time in [7] and [9]. The authors considered the case of the Gibbs-Schr{\"o}dinger semigroups. They scrutinised in these papers a dependence of the rate of convergence for the (exponential) Trotter formula on the smoothness of the potential in the Schr{\"o}dinger generator. The first abstract result in this direction was due to [19]. In this paper a general scheme of lifting the operator-norm rate convergence for the Trotter-Kato product formulae was proposed and advocated for estimation the rate of the trace-nor

Croatia Insurance Building in Zagreb: Research into Architectural Shaping with High Technology

Poslovna zgrada Croatia osiguranja u Zagrebu arhitekta Velimira Neidhardta među najvaćnijim je arhitektonskim ostvarenjima zagrebačke arhitekture na početku 21. stoljeća. Interpolirana s kasnomodernističkom poslovnom zgradom Lloyda arhitekta Marjana Haberlea, zgrada Croatia osiguranja u sebi uspješno spaja tradiciju domaće moderne i postmoderne sa suvremenim arhitektonskim zbivanjima.The offices of the Croatia Insurance Building in Zagreb by architect Velimir Neidhardt is one of the most significant architectural creations of architecture in Zagreb at the beginning of the 21st century. Interpolated with Lloyd\u27s late-modern office building by architect Marijan Haberle, the Croatia Insurance building within itself successfully joins the tradition of domestic modern and postmodern with modern architectural happenings

Trotter-Kato product formula for unitary groups

Let $A$ and $B$ be non-negative self-adjoint operators in a separable Hilbert space such that its form sum $C$ is densely defined. It is shown that the Trotter product formula holds for imaginary times in the $L^2$-norm, that is, one has % % \begin{displaymath} \lim_{n\to+\infty}\int^T_0 \|(e^{-itA/n}e^{-itB/n})^nh - e^{-itC}h\|^2dt = 0 \end{displaymath} % % for any element $h$ of the Hilbert space and any $T > 0$. The result remains true for the Trotter-Kato product formula % % \begin{displaymath} \lim_{n\to+\infty}\int^T_0 \|(f(itA/n)g(itB/n))^nh - e^{-itC}h\|^2dt = 0 \end{displaymath} % % where $f(\cdot)$ and $g(\cdot)$ are so-called holomorphic Kato functions; we also derive a canonical representation for any function of this class

On the unitary equivalence of absolutely continuous parts of self-adjoint extensions

The classical Weyl-von Neumann theorem states that for any self-adjoint operator $A$ in a separable Hilbert space $\mathfrak H$ there exists a (non-unique) Hilbert-Schmidt operator $C = C^*$ such that the perturbed operator $A+C$ has purely point spectrum. We are interesting whether this result remains valid for non-additive perturbations by considering self-adjoint extensions of a given densely defined symmetric operator $A$ in $\mathfrak H$ and fixing an extension $A_0 = A_0^*$. We show that for a wide class of symmetric operators the absolutely continuous parts of extensions $\widetilde A = {\widetilde A}^*$ and $A_0$ are unitarily equivalent provided that their resolvent difference is a compact operator. Namely, we show that this is true whenever the Weyl function $M(\cdot)$ of a pair $\{A,A_0\}$ admits bounded limits M(t) := \wlim_{y\to+0}M(t+iy) for a.e. $t \in \mathbb{R}$. This result is applied to direct sums of symmetric operators and Sturm-Liouville operators with operator potentials

The upcoming New Pact on Migration and Asylum: Will it be up to the challenge? EPC Discussion Paper 29 April 2020

A courageous and ambitious New Pact on Migration and Asylum is one that strengthens the right to asylum; sets the conditions for more equal relationships with third countries when it comes to managing migration; and puts forward a mechanism that can foster genuine solidarity between member states. When the new Commission entered into office in December 2019, it promised a fresh start on migration. President Ursula von der Leyen pledged to deliver a ‘New Pact’ which would break the deadlock between member states on long-awaited reforms, striking a more equitable balance between solidarity and responsibility

A trace formula for functions of contractions and analytic operator Lipschitz functions

In this note we study the problem of evaluating the trace of $f(T)-f(R)$, where $T$ and $R$ are contractions on Hilbert space with trace class difference, i.e., $T-R\in\boldsymbol{S}_1$ and $f$ is a function analytic in the unit disk ${\Bbb D}$. It is well known that if $f$ is an operator Lipschitz function analytic in ${\Bbb D}$, then $f(T)-f(R)\in\boldsymbol{S}_1$. The main result of the note says that there exists a function $\boldsymbol{\xi}$ (a spectral shift function) on the unit circle ${\Bbb T}$ of class $L^1({\Bbb T})$ such that the following trace formula holds: $\operatorname{trace}(f(T)-f(R))=\int_{\Bbb T} f'(\zeta)\boldsymbol{\xi}(\zeta)\,d\zeta$, whenever $T$ and $R$ are contractions with $T-R\in\boldsymbol{S}_1$ and $f$ is an operator Lipschitz function analytic in ${\Bbb D}$.Comment: 6 page

Scattering matrices and Weyl functions

For a scattering system $\{A_\Theta,A_0\}$ consisting of selfadjoint extensions $A_\Theta$ and $A_0$ of a symmetric operator $A$ with finite deficiency indices, the scattering matrix \{S_\gT(\gl)\} and a spectral shift function $\xi_\Theta$ are calculated in terms of the Weyl function associated with the boundary triplet for $A^*$ and a simple proof of the Krein-Birman formula is given. The results are applied to singular Sturm-Liouville operators with scalar and matrix potentials, to Dirac operators and to Schr\"odinger operators with point interactions.Comment: 39 page