Spectral operators generated by damped hyperbolic equations.

*(English)*Zbl 0912.47016The author presents results on the spectral analysis of two nonselfadjoint operators which are the dynamics generators for systems governed by hyperbolic equations containing dissipative terms, namely, the nonhomogeneous damped string and the 3-dimensional damped wave equation with spacially nonhomogeneous spherically symmetric coefficients. Nonselfadjoint boundary conditions are imposed at the ends of a finite interval or on a sphere centered at the origin, respectively. It is shown that these operators are spectral in the sense of N. Dunford using the fact that the systems of root vectors form a Riesz basis in the corresponding energy spaces. Asymptotics of the spectra are obtained and results are stated on the property of the Riesz bases for the nonselfadjoint operator pencils associated with these operators.

Reviewer: R.Vaillancourt (Ottawa)

##### MSC:

47B40 | Spectral operators, decomposable operators, well-bounded operators, etc. |

47A56 | Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) |

35L10 | Second-order hyperbolic equations |

##### Keywords:

spectral operators; damped hyperbolic equations; asymptotics of spectra; operator pencils; spectral analysis; nonselfadjoint operators; specially nonhomogeneous spherically symmetric coefficients; nonselfadjoint boundary conditions; Riesz bases; energy spaces
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\textit{M. A. Shubov}, Integral Equations Oper. Theory 28, No. 3, 358--372 (1997; Zbl 0912.47016)

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