M. V. Markin

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Articles: 8

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

Marat V. Markin

↓ Abstract   |   Article (.pdf)

MFAT 24 (2018), no. 4, 349-369

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

Marat V. Markin

↓ Abstract   |   Article (.pdf)

MFAT 24 (2018), no. 1, 53-70

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

Marat V. Markin

↓ Abstract   |   Article (.pdf)

MFAT 23 (2017), no. 1, 60-65

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

Marat V. Markin

↓ Abstract   |   Article (.pdf)

MFAT 22 (2016), no. 2, 169-183

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

Marat V. Markin

↓ Abstract   |   Article (.pdf)

MFAT 21 (2015), no. 4, 361-369

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

M. V. Markin

↓ Abstract   |   Article (.pdf)

MFAT 12 (2006), no. 3, 254-257

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

Marat V. Markin

MFAT 11 (2005), no. 1, 63-64

63-64

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

M. V. Markin

MFAT 2 (1996), no. 2, 86-93

86-93


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