Spectral theory of regular quasiexponentials and regular \(B\)-representable vector-valued functions (the projection method: 20 years later) (Q2761483)

The starting point for the present paper lies in the work of \textit{S. V. Khrushchev, N. K. Nikolskij} and \textit{B. S. Pavlov}, who considered the problem of when a family of exponentials NEWLINE\[NEWLINE \{e^{i\lambda_kt}: \lambda_k\in \Lambda,\;\inf \operatorname {Im}\lambda _k>0\} NEWLINE\]NEWLINE forms an unconditional basis in \(L^2[0,a]\) [cf. Lecture Notes Math. 864, 214-335, Berlin (1981; Zbl 0466.46018)]. NEWLINENEWLINENEWLINEMore generally, an entire function \(e:\mathbb{C}\to {\mathcal H}\), where \({\mathcal H}\) is a separable Hilbert space, is called a quasi-exponential if: (a) for each \(z\in \mathbb{C}\) there exists a vector \(h_z\in {\mathcal H}\), such that NEWLINE\[NEWLINE\bigl({\widetilde e}(\lambda), e(z)\bigr)=\bigl(e(\lambda),h_z\bigr),\quad {\widetilde e}(z)=z^{-1}\bigl(e(z)-e(0)\bigr),\quad \lambda\in\mathbb{C},NEWLINE\]NEWLINE (b) \(\text{Im}(\sum_k c_k {\widetilde e} (\lambda _k) , \sum_k c_k e(\lambda _k)) \geq 0.\) NEWLINENEWLINENEWLINEThis paper surveys, with some new results as well, the question of when a family of vectors \(\{e(\lambda_k)\}\) generated by a quasiexponential forms an unconditional basis of \({\mathcal H}\). Specifically, a quasi-exponential \(e\) is said to be \(d\)-regular if there exists a sequence \(\{\lambda _k\}^{\infty}_{-\infty}\) with \(\inf_k\operatorname {Im}\lambda _k>-\infty \) such that the family \(\{e(\lambda _k)\}^{+\infty}_{-\infty}\) forms an unconditional basis in \({\mathcal H}\). Also, \(e\) is said to be \(c\)-regular if there are constants \(m, M>0\) such that NEWLINE\[NEWLINEm\|h\|^2\leq \int _{\mathbf R} |(e(x),h)|^2 \|e(x)\|^{-2} dx \leq M\|h\|^2.NEWLINE\]NEWLINE For examples of quasiexponentials, consider the class of operators \(B\) for which \(\sigma(B)=\{0\}\), \(\ker(B)=\{0\}\), and \((B-B^*)/{2i}\geq 0\), and where the resolvent \((I-zB)^{-1}\) is an entire function of exponential type. For such an operator \(B\) together with a fixed element \(g\in{\mathcal H}\), the entire function \(e(z)=(I-zB)^{-1}g\) is a quasiexponential. Taking \(B=J_a:f\mapsto \int^t_0f(s) ds\) to be the integration operator on \(L^2[0,a]\) and \(g\) in \(L^2[0,a]\), we get \(e(z)=(I-zJ_a)^{-1}g = {{d}\over{dt}}\int_{0}^{t} g(t-s)e^{izs} ds\). In section 5, the author describes all regular quasiexponentials of this type. For instance, letting \(g=\Gamma^{-1}(\alpha)t^{\alpha -1}\) \((\alpha >1/2)\) yields the Mittag-Leffler function \(e(z)=t^{\alpha -1}E_1(it,\alpha)\), where \(E_1(z,\alpha)=\sum_{k=0}^{\infty}{{\Gamma}^{-1}(\alpha+k)z^k}\). (In particular, \(e(z)=\exp(izt)\) if \(\alpha =1\).) This function is regular only for \(1/2 <\alpha < 3/2\). An important subclass of the quasiexponentials just described is as follows. Let \(w^2\) be an arbitrary weight on \(\mathbb{R}\) satisfying the \(A_2\)-Muckenhoupt condition. Lemma 2.1 shows that there is an outer function \(w_-\) defined on the domain \(\text{ Im} z<0\) such that \(|w_-(x-i0)|^2=w^2(x)\) for a.e. \(x \in \mathbb{R}\) and such that NEWLINE\[NEWLINEw_-(z)=z\int ^{\infty}_0e^{-izt}y_w(t) dt, \quad \text{Im} z <0NEWLINE\]NEWLINE for some function \(y_w\in L^2_{\text{loc}}(\mathbb{R}_+)\). This in turn is shown to generate the quasiexponential \(e_w\), with values in \(L^2[0,a]\), defined by setting NEWLINE\[NEWLINEe_w(z,t):= {{d}\over{dt}}\int^t_0y_w(t-s)e^{izs} ds.NEWLINE\]NEWLINE The Mittag-Leffler function \(E_1(z,\alpha)\) coincides with the quasiexponential \(e_w\) generated by the weight \(w^2(x)=|x|^{2(1-\alpha)}.\) Theorem 2.10 shows that all such quasiexponentials are regular. These are but a few of the paper's highlights.