Bases in Hilbert Spaces

Abstract

This chapter tries to collect some basic facts for Riesz basis with simple introduction nature. These facts are minimal for understanding the applications in later chapters. These include basis property of exponential family $e^{i\lambda _{n_{t}}}$, which is also referred to as nonharmonic Fourier series; the property of entire functions, specially for entire function of exponential type; the materials about the Hardy space and Blaschke are only limited to a few results; The D-type operator and One-rank perturbation for D-type operator are also introduced. The materials of generalized divided difference (GDD) developed by Russian school is also introduced. The Pavlov theorem and Keldysh theorem are specially introduced. It also presents the general results on the Riesz basis property for $C_0$-groups and semigroups

Remarks and Bibliographical Notes

This chapter tries to collect some basic facts for Riesz basis with simple introduction nature. These facts are minimal for understanding the applications in later chapters. For basis property of exponential family $\{e^{i\lambda _n t}\}$, which is also referred to as nonharmonic Fourier series, the standard reference book is Young (2001), Avodonin and Ivanov (1995) and Nikol’skiĭ (1986). The book Avodonin and Ivanov (1995) is more about families of vector-valued exponentials and applications to controllability on systems described by partial differential equations. For the property of entire functions, specially for entire function of exponential type, a nice reference book is Levin (1980) but some of the results can be found in Young (2001), Titchmarsh (1953). For unconditional bases in Banach spaces, the classical book is Singer (1970). Most of the materials about the Hardy space and Blaschke product can be found from Koosis (1998); Garnett (1981) but the proofs in this chapter are only limited to a few results and a couple of results have to be referred to Koosis (1998), Garnett (1981). The D-type operator was first introduced in Li (1978) and one-rank perturbation for D-type operator was introduced from Guo and Luo (2002) ($\copyright$[2002] IEEE. Reprinted, with permission, from Guo and Luo 2002). For the materials of GDD, we refer to Avdonin and Ivanov (2002). Theorem 2.9 was first introduced in Rao (1997). The Pavlov theorem is completely rewritten in a different way and further reading is referred to Pavlov (1979), Hruščëv et al. (1981), and Avodonin and Ivanov (1995). Lemma 2.31 and Theorem 2.47 come from Xu and Guo (2003) ($\copyright$[2003] SIAM. Reprinted, with permission, from Xu and Guo 2003). Lemma 2.21 comes from Dunford and Schwartz (1971) and more discussions on the Ingham theorem 2.21 and its applications can be found in Komornik and Loreti (2005). Theorem 2.48 is taken from Guo and Xu (2006) ($\copyright$[2006] Elsevier. Reprinted, with permission, from Guo and Xu 2006). Corollary 2.7 was obtained in Zwart et al. (2010) by interpolation function method. Theorem 2.22 and Proposition 2.5 were first obtained in Shkalikov (1986). For Keldysh theorem 2.50 and related introduction, we refer to Gohberg et al. (1990). The formula (2.258) was introduced in Arendt et al. (1986, p. 73).