We consider nonlinear equations in linear spaces and algebras which can be solved by a "separation of variables" obtained due to Algebraic Analysis. It is shown that the structures of linear spaces and commutative algebras (even if they are Leibniz algebras) are not rich enough for our purposes. Therefore, in order to generalize the method used for separable ordinary differential equations, we have to assume that in algebras under consideration there exist logarithmic mappings. Section 1 contains...

In this work we introduce a nonparametric recursive aggregation process called Multilayer Aggregation (MLA). The name refers to the fact that at each step the results from the previous one are aggregated and thus, before the final result is derived, the initial values are subjected to several layers of aggregation. Most of the conventional aggregation operators, as for instance weighted mean, combine numerical values according to a vector of weights (parameters). Alternatively, the MLA operators...

We study the frequency of hypercyclicity of hypercyclic, non–weakly mixing linear operators. In particular, we show that on the space ${\ell }^1\left(\mathbb{N}\right)$, any sublinear frequency can be realized by a non–weakly mixing operator. A weaker but similar result is obtained for $c_0\left(\mathbb{N}\right)$ or ${\ell }^p\left(\mathbb{N}\right)$, $1\<p\<\infty$. Part of our results is related to some Sidon-type lacunarity properties for sequences of natural numbers.

In this note we discuss some results on numerical radius attaining operators paralleling earlier results on norm attaining operators. For arbitrary Banach spaces X and Y, the set of (bounded, linear) operators from X to Y whose adjoints attain their norms is norm-dense in the space of all operators. This theorem, due to W. Zizler, improves an earlier result by J. Lindenstrauss on the denseness of operators whose second adjoints attain their norms, and is also related to a recent result by C. Stegall...

We show that a positive semigroup $T_t$ on $L_p(\Omega ,\nu )$ with generator A and ||R(α + i β)|| → 0 as |β| → ∞ for some α ∈ ℝ is continuous in the operator norm for t>0. The proof is based on a criterion for norm continuity in terms of “smoothing properties” of certain convolution operators on general Banach spaces and an extrapolation result for the $L_p$-scale, which may be of independent interest.

Let $f$ be a continuous function on $I$ and $A$, $B\in {\mathrm{𝒮𝒜}}_I\left(H\right)$, the convex set of selfadjoint operators with spectra in $I$. If $A\ne B$ and $f$, as an operator function, is Gateaux differentiable on $$[A,B]:=\left\{(1-t)A+tB\mid t\in \left[0,1\right]\right\},$$ while $p:\left[0,1\right]\to ℝ$ is Lebesgue integrable, then we have the inequalities $$\begin{array}{cc}\hfill \parallel {\int }_0^{1}p\left(\tau \right)& f\left(\left(1-\tau \right)A+\tau B\right)d\tau -{\int }_0^{1}p\left(\tau \right)d\tau {\int }_0^{1}f\left(\left(1-\tau \right)A+\tau B\right)d\tau \parallel \hfill \\ & \le {\int }_0^1\tau (1-\tau )|\frac{{\int }_{\tau }^{1}p\left(s\right)ds}{1-\tau }-\frac{{\int }_0^{\tau }p\left(s\right)ds}{\tau }|∥\nabla f_{\left(1-\tau \right)A+\tau B}\left(B-A\right)∥d\tau \hfill \\ & \le \frac{1}{4}{\int }_0^1|\frac{{\int }_{\tau }^{1}p\left(s\right)ds}{1-\tau }-\frac{{\int }_0^{\tau }p\left(s\right)ds}{\tau }|∥\nabla f_{\left(1-\tau \right)A+\tau B}\left(B-A\right)∥d\tau ,\hfill \end{array}$$ where $\nabla f$ is the Gateaux derivative of $f$.