# M. V. Markin

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### On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator

Methods Funct. Anal. Topology **24** (2018), no. 4, 349-369

Found are conditions on a scalar type spectral operator $A$ in a complex Banach space necessary and sufficient for all weak solutions of the evolution equation \begin{equation*} y'(t)=Ay(t),\quad t\ge 0, \end{equation*} to be strongly Gevrey ultradifferentiable of order $\beta\ge 1$, in particular analytic or entire, on $[0,\infty)$. Certain inherent smoothness improvement effects are analyzed.

### On the mean ergodicity of weak solutions of an abstract evolution equation

Methods Funct. Anal. Topology **24** (2018), no. 1, 53-70

Found are conditions of rather general nature sufficient for the existence of the limit at infinity of the Cesàro means $$ \frac{1}{t} \int_0^ty(s)\,ds $$ for every bounded weak solution $y(\cdot)$ of the abstract evolution equation $$ y'(t)=Ay(t),\ t\ge 0, $$ with a closed linear operator $A$ in a Banach space $X$.

### On certain spectral features inherent to scalar type spectral operators

Methods Funct. Anal. Topology **23** (2017), no. 1, 60-65

Important spectral features such as the emptiness of the residual spectrum, countability of the point spectrum, provided the space is separable, and a characterization of spectral gap at 0, known to hold for bounded scalar type spectral operators, are shown to naturally transfer to the unbounded case.

### On the generation of Beurling type Carleman ultradifferentiable $C_0$-semigroups by scalar type spectral operators

Methods Funct. Anal. Topology **22** (2016), no. 2, 169-183

A characterization of the scalar type spectral generators of Beurling type Carleman ultradifferentiable $C_0$-semigroups is established, the important case of the Gevrey ultradifferentiability is considered in detail, the implementation of the general criterion corresponding to a certain rapidly growing defining sequence is observed.

### On the Carleman ultradifferentiable vectors of a scalar type spectral operator

Methods Funct. Anal. Topology **21** (2015), no. 4, 361-369

A description of the Carleman classes of vectors, in particular the Gevrey classes, of a scalar type spectral operator in a reflexive complex Banach space is shown to remain true without the reflexivity requirement. A similar nature description of the entire vectors of exponential type, known for a normal operator in a complex Hilbert space, is generalized to the case of a scalar type spectral operator in a complex Banach space.

### A note on one decomposition of Banach spaces

Methods Funct. Anal. Topology **12** (2006), no. 3, 254-257

For a scalar type spectral operator $A$ in complex Banach space $X$, the decomposition of $X$ into the direct sum \begin{equation*} X=\ker A\oplus \overline{R(A)}, \end{equation*} where $\ker A$ is the kernel of $A$ and $\overline{R(A)}$ is the closure of its range $R(A)$ is established.

### A Gelfand-Mazur type theorem for normed algebras

Methods Funct. Anal. Topology **11** (2005), no. 1, 63-64

### On a characterization of the generators of analytic semigroups in the cass of normal operators

Methods Funct. Anal. Topology **2** (1996), no. 2, 86-93