Abstract:
The author studies the spectral properties of the operator $A=i\frac d{dt}$ in the space $L_2(0,1)$, whose domain of definition is the kernel of some functional that is bounded in $W_2^1(0,1)$ but not bounded in $L_2(0,1)$. Necessary and sufficient conditions are given under which the operators $\pm iA$ generate $C_0$-semigroups, and criteria for the similarity of $A$ with a dissipative operator are proved. The results are used to study the basis properties of families of exponentials and to solve S. G. Krein's problem on the description of generators of semigroups in terms of their dissipative extensions. The solvability of integral equations of Delsarte type for mean periodic extensions of functions is also proved.
Bibliography: 32 titles.
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