##
**Almost automorphic and almost periodic functions in abstract spaces.**
*(English)*
Zbl 1001.43001

New York, NY: Kluwer Academic/Plenum Publishers. x, 138 p. (2001).

This book should be doubly welcome: Almost automorphic (aa) functions and solutions of differential equations have not up till now appeared in book form, and contrary to the now usual monographs of 400 plus pages the author manages to do with only 138. For \(f: \mathbb{R}\to X=\) Banach space, or locally convex, aa, weakly aa, asymptotically aa (on \(\mathbb{R}_+\)) and almost periodic (ap) \(f\) are introduced; for aa Bochner’s definition is used, i.e. to each sequence \((t_n)\) there exists a subsequence \((s_n)\) such that \(\lim_n f(t+ s_n)\) exists for each \(t\in \mathbb{R}\), \(:= g(t)\), and \(\lim_n g(t- s_n)= f(t)\) for each \(t\in\mathbb{R}\), with pointwise limits only (uniform limits characterize ap), and \(f\) continuous.

The author then gives the Bohl-Bohr-Bochner-Amerio results on indefinite integrals of aa and ap functions (but no harmonic analysis or synthesis): If \(f: \mathbb{R}\to X\) is aa and \(Pf(t):= \int^t_0 f(s) ds\), then \(Pf\) is aa if and only if \(Pf\) has relatively compact (rc) range; if \(X\) is uniformly convex, “rc” can be replaced by “bounded”; similarly for ap.

Then aa and ap solutions of \((*)\) \(x'= Ax+ f\) on \(\mathbb{R}\) are treated, with \(A=\) infinitesimal generator of a one-parameter \(C_0\)-semigroup of operators \(T(t): X\to X\). Samples: If \((T(t))\) is a \(C_0\)-group and \(f\) and \(Tx\) are aa for all \(x\in X\), then every mild solution of \((*)\) with rc range is aa; if \(X\) is uniformly convex, “bounded” instead of rc suffices. If \((T(t))\) is only a \(C_0\)-semigroup, the same holds if additionally \(T(t) x\to 0\) as \(t\to\infty\) for all \(x\in X\). Similar results hold for ap instead of aa, provided \((T(t))_{t\in\mathbb{R}}\) is equicontinuous. If \(A\) is a symmetric operator on \(X=\) Hilbert space, then any solution of \(x'= Ax\) with rc range is ap. Also linear finite-dimensional or compact or suitably nonlinear Lipschitz-perturbations of \((*)\) are considered. A correspondence between aa solutions of \((*)\) and asymptotically aa solutions of a certain nonlinear differential equation is established, furthermore dynamical systems associated with \(C_0\)-semigroups are discussed.

There are absolutely no examples, or remarks discussing the various assumptions. The exposition is not as clear, precise or selfcontained as one might wish: Often earlier results are used implicitly without stating so and especially without citing the exact lemma or theorem involved, which is hard on the reader; furthermore, frequently non-trivial results of functional analysis are used or stated without giving any references (or assumptions).

Proposition 3.2.10, Theorem 4.1.2 and Theorem 4.2.1 are obviously false as stated. If \(g: \mathbb{R}\to X\) has rc range, the indefinite integral of \(g\) need not have rc range, as used on p. 110, line 3. There are numerous misprints and (notational) inconsistencies. The index (less than one page) is of little help, e.g. for “perfect Banach space” it refers to p. 48, where nothing can be found; here the author does not seem to remember that he gave two different definitions for “perfect” (p. 40, Def. 2.6.1, pp. 103/104, Def. 7.1.7), and he seems to be unaware of the fact that definition 7.1.7 characterizes \(X\) not containing \(c_0\) [M. I. Kadets, Funct. Anal. Appl. 3, 228-230 (1969; Zbl 0206.42903)], and that, at least for uniformly continuous functions, both definitions are equivalent [B. Basit and H. Günzler, J. Diff. Equations 149, 115-142 (1998; Zbl 0912.45013); pp. 119 and 120, and the references there].

So this book cannot be recommended for students or for a lecture (too much additional work needed), but it should be good for a lively seminar.

The author then gives the Bohl-Bohr-Bochner-Amerio results on indefinite integrals of aa and ap functions (but no harmonic analysis or synthesis): If \(f: \mathbb{R}\to X\) is aa and \(Pf(t):= \int^t_0 f(s) ds\), then \(Pf\) is aa if and only if \(Pf\) has relatively compact (rc) range; if \(X\) is uniformly convex, “rc” can be replaced by “bounded”; similarly for ap.

Then aa and ap solutions of \((*)\) \(x'= Ax+ f\) on \(\mathbb{R}\) are treated, with \(A=\) infinitesimal generator of a one-parameter \(C_0\)-semigroup of operators \(T(t): X\to X\). Samples: If \((T(t))\) is a \(C_0\)-group and \(f\) and \(Tx\) are aa for all \(x\in X\), then every mild solution of \((*)\) with rc range is aa; if \(X\) is uniformly convex, “bounded” instead of rc suffices. If \((T(t))\) is only a \(C_0\)-semigroup, the same holds if additionally \(T(t) x\to 0\) as \(t\to\infty\) for all \(x\in X\). Similar results hold for ap instead of aa, provided \((T(t))_{t\in\mathbb{R}}\) is equicontinuous. If \(A\) is a symmetric operator on \(X=\) Hilbert space, then any solution of \(x'= Ax\) with rc range is ap. Also linear finite-dimensional or compact or suitably nonlinear Lipschitz-perturbations of \((*)\) are considered. A correspondence between aa solutions of \((*)\) and asymptotically aa solutions of a certain nonlinear differential equation is established, furthermore dynamical systems associated with \(C_0\)-semigroups are discussed.

There are absolutely no examples, or remarks discussing the various assumptions. The exposition is not as clear, precise or selfcontained as one might wish: Often earlier results are used implicitly without stating so and especially without citing the exact lemma or theorem involved, which is hard on the reader; furthermore, frequently non-trivial results of functional analysis are used or stated without giving any references (or assumptions).

Proposition 3.2.10, Theorem 4.1.2 and Theorem 4.2.1 are obviously false as stated. If \(g: \mathbb{R}\to X\) has rc range, the indefinite integral of \(g\) need not have rc range, as used on p. 110, line 3. There are numerous misprints and (notational) inconsistencies. The index (less than one page) is of little help, e.g. for “perfect Banach space” it refers to p. 48, where nothing can be found; here the author does not seem to remember that he gave two different definitions for “perfect” (p. 40, Def. 2.6.1, pp. 103/104, Def. 7.1.7), and he seems to be unaware of the fact that definition 7.1.7 characterizes \(X\) not containing \(c_0\) [M. I. Kadets, Funct. Anal. Appl. 3, 228-230 (1969; Zbl 0206.42903)], and that, at least for uniformly continuous functions, both definitions are equivalent [B. Basit and H. Günzler, J. Diff. Equations 149, 115-142 (1998; Zbl 0912.45013); pp. 119 and 120, and the references there].

So this book cannot be recommended for students or for a lecture (too much additional work needed), but it should be good for a lively seminar.

Reviewer: Hans F.Günzler (Kiel)

### MSC:

43-02 | Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis |

34C27 | Almost and pseudo-almost periodic solutions to ordinary differential equations |

34G10 | Linear differential equations in abstract spaces |

43A60 | Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions |

47D06 | One-parameter semigroups and linear evolution equations |