<p>In this paper, we consider the application of the method of equivalent operators to the differential operator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {L}=-d/dt+A$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">L</mi> <mo>=</mo> <mo>-</mo> <mi>d</mi> <mo stretchy="false">/</mo> <mi>d</mi> <mi>t</mi> <mo>+</mo> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation> with the domain <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$D(\mathcal {L})\subset \mathcal {F}\rightarrow \mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">L</mi> <mo stretchy="false">)</mo> <mo>⊂</mo> <mi mathvariant="script">F</mi> <mo stretchy="false">→</mo> <mi mathvariant="script">F</mi> </mrow> </math></EquationSource> </InlineEquation> acting in a homogeneous space of functions <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation>. We assume that the operator <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$A: D(A)\subset \mathcal {H}\rightarrow \mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>:</mo> <mi>D</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>⊂</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">→</mo> <mi mathvariant="script">H</mi> </mrow> </math></EquationSource> </InlineEquation> is a normal operator with compact resolvent in the Hilbert space <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>. Conditions for its invertibility and estimates for the norm of the inverse in various spaces <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation> are given.</p>

THE METHOD OF EQUIVALENT OPERATORS IN THE STUDY OF ONE CLASS OF FIRST-ORDER DIFFERENTIAL OPERATORS WITH CONSTANT OPERATOR COEFFICIENTS

期刊论文
发表日期：

2025年10月1日

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A. G. Baskakov,
G. V. Garkavenko,
L. N. Kostina,
N. B. Uskova

摘要

In this paper, we consider the application of the method of equivalent operators to the differential operator $\mathcal {L}=-d/dt+A$ $L = - d / d t + A$ with the domain $D(\mathcal {L})\subset \mathcal {F}\rightarrow \mathcal {F}$ $D (L) \subset F \to F$ acting in a homogeneous space of functions $\mathcal {F}$ $F$ . We assume that the operator $A: D(A)\subset \mathcal {H}\rightarrow \mathcal {H}$ $A : D (A) \subset H \to H$ is a normal operator with compact resolvent in the Hilbert space $\mathcal {H}$ $H$ . Conditions for its invertibility and estimates for the norm of the inverse in various spaces $\mathcal {F}$ $F$ are given.