Abstract:
We investigate the boundedness of the $H^\infty$-calculus by estimating the bound $b(\varepsilon)$ of the mapping $H^{\infty}\rightarrow \mathcal{B}(X)$: $f\mapsto f(A)T(\varepsilon)$ for $\varepsilon$ near zero. Here, $-A$ generates the analytic semigroup $T$ and $H^{\infty}$ is the space of bounded analytic functions on a domain strictly containing the spectrum of $A$. We show that $b(\varepsilon)=\mathcal{O}(|\log\varepsilon|)$ in general, whereas $b(\varepsilon)=\mathcal{O}(1)$ for bounded calculi. This generalizes a result by Vitse and complements work by Haase and Rozendaal for non-analytic semigroups. We discuss the sharpness of our bounds and show that single square function estimates yield $b(\varepsilon)=\mathcal{O}(\sqrt{|\log\varepsilon|})$.

Abstract:
We show $H^{\infty}$-functional calculus estimates for Tadmor-Ritt operators (also known as Ritt operators), which generalize and improve results by Vitse. These estimates are in conformity with the best known power-bounds for Tadmor-Ritt operators in terms of the constant dependence. Furthermore, it is shown how discrete square function estimates influence the estimates.

Abstract:
We show that, given a Banach space and a generator of an exponentially stable $C_{0}$-semigroup, a weakly admissible operator $g(A)$ can be defined for any $g$ bounded, analytic function on the left half-plane. This yields an (unbounded) functional calculus. The construction uses a Toeplitz operator and is motivated by system theory. In separable Hilbert spaces, we even get admissibility. Furthermore, it is investigated when a bounded calculus can be guaranteed. For this we introduce the new notion of exact observability by direction. Finally, it is shown that the calculus coincides with one for half-plane-operators.

Abstract:
Assume that a block operator of the form $\left(\begin{smallmatrix}A_{1}\\A_{2}\quad 0\end{smallmatrix}\right)$, acting on the Banach space $X_{1}\times X_{2}$, generates a contraction $C_{0}$-semigroup. We show that the operator $A_{S}$ defined by $A_{S}x=A_{1}\left(\begin{smallmatrix}x\\SA_{2}x\end{smallmatrix}\right)$ with the natural domain generates a contraction semigroup on $X_{1}$. Here, $S$ is a boundedly invertible operator for which $\epsilon\ide-S^{-1}$ is dissipative for some $\epsilon>0$. With this result the existence and uniqueness of solutions of the heat equation can be derived from the wave equation.

Abstract:
In this short note we use ideas from systems theory to define a functional calculus for infinitesimal generators of strongly continuous semigroups on a Hilbert space. Among others, we show how this leads to new proofs of (known) results in functional calculus.

Abstract:
In this paper we show that from the estimate $\sup_{t \geq 0}\|C(t) - \cos(at)I\| <1$ we can conclude that $C(t)$ equals $\cos(at) I$. Here $\left(C(t)\right)_{t \geq 0}$ is a strongly continuous cosine family on a Banach space.

Abstract:
For $\left(C(t)\right)_{t \geq 0}$ being a strongly continuous cosine family on a Banach space, we show that the estimate $\limsup_{t\to 0^{+}}\|C(t) - I\| <2$ implies that $C(t)$ converges to $I$ in the operator norm. This implication has become known as the zero-two law. We further prove that the stronger assumption of $\sup_{t\geq0}\|C(t)-I\|<2$ yields that $C(t)=I$ for all $t\geq0$. Additionally, we derive alternative proofs for similar results for $C_{0}$-semigroups.

Abstract:
For $\left(C(t)\right)_{t\in\mathbb R}$ being a cosine family on a unital normed algebra, we show that the estimate $\limsup_{t\to\infty^{+}}\|C(t) - I\| <2$ implies that $C(t)=I$ for all $t\in\mathbb R$. This generalizes the result that $\sup_{t\geq0}\|C(t)-I\|<2$ yields that $C(t)=I$ for all $t\geq0$. We also state the corresponding result for discrete cosine families and for semigroups.

Abstract:
The phenomenon of the cosmological acceleration discovered in 1998 is usually explained as a manifestation of a hypothetical field called dark energy which is believed to contain more than 70% of the energy of the Universe. This explanation is based on the assumption that empty space-time background should be flat and hence a nonzero curvature of the background is a manifestation of a hidden matter. We argue that quantum theory should proceed not from space-time background but from a symmetry algebra. Then the cosmological acceleration can be easily and naturally explained from first principles of quantum theory without involving empty space-time background, dark energy and other artificial notions. We do not assume that the reader is an expert in the given field and the content of the paper can be understood by a wide audience of physicists.

Abstract:
We clarify the relation between the perception of narratives and aesthetic emotions by relating them to mechanisms of knowledge-acquisition. Stories elicit emotions by diverging from expectations one may formulate on the basis of their properties. The greater the divergence, the stronger the emotion. Models of emotions, expectations, and knowledge-acquisition are briefly presented. We relate them to research pertaining to narrative structures and provide a mathematical description for aesthetic emotions. We conclude by underlining the fundamental role played by aesthetic emotions in the workings of the human mind.