Marianna A. Shubov
7 PROPERTIES OF THE ADJOINT OPERATORS AND OPERATOR-VALUED FUNCTIONS
To write the spectral decomposition for the generalized resolvent operator, we need detailed information about the set of functions that is biorthogonal to the set of the mode shapes. It turns out that this biorthogonal set is closely related to the set of nontrivial solutions of the following equation:
S∗(λ)Φ ≡ [−iL∗βδ+ M∗+ N∗(λ)]Φ = λΦ. (7.1) The set of complex pointsλ for which Eq. (7.1) has nontrivial solutions will be called the set of theadjoint aeroelastic modes and the set of corresponding solutions will be called the set ofadjoint mode shapes.
Now we describe the structure of the adjoint analytic operator-valued function S∗(λ). We already know that L∗βδ is given by the same matrix differential expression (3.6) and the only difference is in the description of the domain ofL∗βδ, i.e., the parameters β and δ from (3.7) should be replaced with (− ¯β) and (−¯δ) respectively.
Theorem 7.1. The operator-valued function M∗+ N∗(λ) can be represented in the following form:
πρ/G. If u satisfie the second condition from (2.14), then W (ω) = 0.
Theorem 7.2. The set of adjoint aeroelastic modes is countable and does not have accumulation points on the complex place C. This set splits asymptotically into two series, which we call the β∗-branch and the δ∗-branch. Asymptotical distribution of the β∗ and the δ∗-branches of the adjoint aeroelastic modes can be obtained from the asymptotical distribution of the spectrum of the operator Lβδ. Namely, if {µβ∗n }n ∈ Z is the β∗-branch of the adjoint aeroelastic modes, then µβ∗n = iˆµβ∗n , and the asymptotics of the set {µβ∗n }n ∈ Z are given by the right-hand side of formula (4.3). Similarly, if {µδ∗n = iˆµδ∗n}n ∈ Z is the δ∗-branch of the adjoint aeroelastic modes, then the asymptotical distribution of the set {µδ∗n } can be obtained from the right-hand side of the formula (4.4) by replacing the sign
“+with the sign “–” before the logarithmic term.
Note that due to the latter theorem, one can see that the set of the adjoint aeroelastic modes is closely related to the spectrum of the operatorL∗βδ.
Finally, we formulate an important result about the operator
Kβδ= Lβδ+ iM. (7.3)
Let us denote by{Φβn}n∈Zand{Φδn}n∈Ztheβ and δ-branches of root vectors of the oper-atorKβδ, respectively.
Theorem 7.3.a) The entire set of the root vectors {Φβn}n∈Z
{Φδn}n∈Zof the oper-ator Kβδ∗ is complete in H. b) This set forms a Riesz basis in H. The properties (a) and (b) imply that the operator Kβδ, is a Riesz spectral operator in the sense of Dunford [22]. c) The operator K∗βδ, which is adjoint to the operator Kβδ, is also Riesz spectral.
To prove Theorem 7.3, we have used the same approach as for the case of the oper-atorLβδ, i.e., the Sz. Nagy–C. Foias functional model for nonself-adjoint operators. To formulate the result on the mode shapes, let us use the following notations: {Fnβ}n∈Zare theβ-branch mode shapes, {Fnδ}n∈Zare theδ-branch mode shapes.
Theorem 7.4.a) The entire set of the mode shapes {Fnβ}n∈Z
{Fnδ}n∈Zis complete in the energy space H. b) The set of mode shapes is quadratically close to the set of root vectors of the operator Kβδ, i.e.,
n∈Z
||Φβn− Fnβ||2H+
n∈Z
||Φδn− Fnδ||2H< ∞. (7.4)
By combining (a) and (b) of Theorem 7.4 with the fact thatK∗βδis a Riesz spectral operator, we show that the set of the mode shapes forms a Riesz basis inH. A similar fact on the Riesz basis property of the set of adjoint mode shapes can be shown as well.
The Riesz basis property of the mode shapes is crucially important for the spectral decomposition of the generalized resolvent operator.
Acknowledgments
Partial support by the National Science Foundation Grants ECS #0080441, DMS#0072247, DMS-9972748, and the Advanced Research Program-97 of Texas Grant 0036-44-045 is highly appreciated.
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