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Lectures on Operator Semigroups

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Lectures on Operator Semigroups Lectures on Operator Semigroups Yuri G. Kondratiev 2017 2 Contents 1 Linear Dynamical Systems5 1.1 Cauchy's Functional Equation...................5 1.2 Finite-Dimensional Systems: Matrix Semigroups.........8 1.3 Uniformly Continuous Operator Semigroups........... 10 1.4 Strongly Continuous Semigroups.................. 11 3 4 Contents Chapter 1 Linear Dynamical Systems There are many good reasons why an "autonomous deterministic system" should be described by maps T ptq; t ¥ 0, satisfying the functional equation (FE) T pt ` sq “ T ptqT psq: Here, t is the time parameter, and each T ptq maps the "state space" of the system into itself. These maps completely determine the time evolution of the system in the following way: If the system is in state x0 at time t0 “ 0 then at time t it is in state T ptqx. However, in most cases a complete knowledge of the maps T ptq is hard, if not impossible, to obtain. It was one of the great discoveries of mathematical physics, based on the invention of calculus, that, as a rule, it is much easier to understand the ”infinitesimal changes" occurring at any given time. In this case, the system can be described by a differential equation. In this chapter we analyze this phenomenon in the mathematical context of linear operators on Banach spaces. For this purpose, we take two opposite views. pV 1q. We start with a solution t ÞÑ T ptq and ask which assumptions imply that it is differentiable and satisfies a differential equation. pV 2q. We start with a differential equation and ask how its solution can be related to a family of mappings. 1.1 Cauchy's Functional Equation As a warm-up, this program will be performed in the scalar-valued case first. In fact, it was A. Cauchy who in 1821 asked in his Cours d'Analyse, without any further motivation, the following question: D'eterminer la fonction φpxq de mani'ere qu'elle reste continue entre deux limites r'eelles quelconques de la variable x, et que l'on ait pour toutes les valeurs r'eelles des variables x et y φpx ` yq “ φpxqφpyq. Using modern notation, we restate his question as follows dropping the continuity requirement for the moment. 5 6 Chapter 1. Linear Dynamical Systems p1:1q Problem. Find all maps T p¨q : R` ÞÑ C satisfying the functional equation T pt ` sq “ T ptqT psqfor all t; s ¥ 0; (1.1) T p0q “ 1: Evidently, the exponential functions t ÞÑ eta satisfy (FE) for any a P C. With his question, Cauchy suggested that these canonical solutions should be all solutions of (FE). Before giving an answer to Problem 1.1, we take a closer look at the exponential functions and observe that they, besides solving the algebraic identity (FE), also enjoy some important analytic properties. Determine the function φpxq in such a way that it remains continuous be- tween two arbitrary real limits of the variable x, and that, for all real values of the variables x and y, one has φpx ` yq “ φpxqφpyq: Proposition 1.1. Let T ptq :“ eta for some a P C all t ¥ 0: Then the function T p¨q is differentiable and satisfies the differential equation (or, more precisely, the initial value problem) (DE) d T ptq “ aT ptq dt for all t ¥ 0, T p0q “ 1. Conversely, the function T p¨q : R` Ñ C defined by T ptq “ eta for some a P C is the only differentiable function satisfying (DE). d Finally, we observe that a “ dtT ptq for t “ 0. Proof. We show only the assertion concerning uniqueness. Let Sp¨q : R` Ñ C be another differentiable function satisfying (DE). Then the new function Qp¨q : r0; ts Ñ C defined by Qpsq :“ T psqSpt ´ sq for 0 ¤ s ¤ t d for some fixed t ¡ 0 is differentiable with derivative dsQpsq ” 0: This shows that T ptq “ Qptq “ Qp0q “ Sptq for arbitrary t ¡ 0. This proposition shows that, in our scalar-valued case, pV 2q can be answered easily using the exponential function. It is now our main point that continuity is already sufficient to obtain differentiability in pV 1q. 1.2. Finite-Dimensional Systems: Matrix Semigroups 7 Proposition 1.2. Let T p¨q : R` Ñ C be a continuous function satisfying (FE). Then T p¨) is differentiable, and there exists a unique a P C such that (DE) holds. Proof. Since T p¨q is continuous on R`, the function V p¨q defined by t V ptq :“ T psqds; t ¥ 0; »0 is differentiable with V 1ptq “ T ptq. This implies V ptq lim “ V 1p0q “ T p0q “ 1: tÑ0` t Therefore, V pt0q is different from zero, hence invertible, for some small t0 ¡ 0. The functional equation (FE) now yields t ´1 ´1 T ptq “ V pt0qV pt0qT ptq “ V pt0q T pt ` sqds »0 t`t0 ´1 ´1 “ V pt0q T psqds “ V pt0qpV pt ` t0q ´ V ptqq »t for all t ¥ 0: Hence, T p¨q is differentiable with derivative d T pt ` hq ´ T ptq “ lim “ dt hÑ0` h T phq ´ T p0q lim T ptq “ T 1p0qT ptq hÑ0` h for all t ¥ 0: This shows that T p¨q satisfies (DE) with a :“ T 1p0q: The combination of both results leads to a satisfactory answer to Cauchy's Problem 1.1. Theorem 1.3. Let T p¨q : R` Ñ C be a continuous function satisfying (FE). Then there exists a unique a P C such that T ptq “ eta for all t ¥ 0. 1.2 Finite-Dimensional Systems: Matrix Semi- groups In this section we pass to a more general setting and consider finite dimensional vector spaces X “ Cn. The space LpXq of all linear operators on X will then be identified with the space MnpCq of all complex n ˆ n matrices, and 8 Chapter 1. Linear Dynamical Systems a linear dynamical system on X will be given by a matrix-valued function T p¨q : R` Ñ MnpCq satisfying the functional equation (FE) T pt ` sq “ T ptqT psq for all t; s ¥ 0;T p0q “ I: As before, the variable t will be interpreted as "time". The "time evolution" of a state x0 P X is then given by the function ξ : R` Ñ X defined as ξptq “ T ptqx0. We also call tT ptqx0 | t ¥ 0u the orbit of x0 under T p¨q. From the functional equation (FE) it follows that an initial state x0 arrives after an elapsed time t ` s at the same state as the initial state y0 :“ T psqx0 after time t. In this new context we study (V1) and (V2) from Section 1 and restate Cauchy's Problem 1.1. Problem. Find all maps T p¨q : R` Ñ MnpCq satisfying the functional equa- tion (FE). Imitating the arguments from Section 1 we first look for "canonical" solu- tions of (FE) and then hope that these exhaust all (natural) linear dynamical systems. As in Section 1 the candidates for solutions of (FE) are the "expo- nential functions". tA Definition 1.4. For any A P MnpCq and t P R the matrix exponential e is defined by 8 tkAk etA :“ : (1.2) k! k`0 ¸ n Taking any norm on C and the corresponding matrix-norm on MnpCq one shows that the partial sums of the series above form a Cauchy sequence; hence the series converges and satisfies }etA} ¤ et}A}: for all t ¥ 0. Moreover, the map t ÞÑ etA has the following properties. Proposition 1.5. For any A P MnpCq the map tA R` Q t Ñ e P MnpCq is continuous and satisfies (FE) ept`sqA “ etAesA for t; ¥ 0; e0A “ I: Proof. Since the series 8 tkAk k! k`0 ¸ 1.2. Finite-Dimensional Systems: Matrix Semigroups 9 converges, one can show, as for the Cauchy product of scalar series, that 8 tkAk 8 skAk k! k! k`0 k`0 ¸ ¸ 8 n tn´kAn´k skAk “ “ n k ! k! n“0 k“0 p ´ q ¸ ¸ 8 pt ` sqnAn : n! n“0 ¸ This proves (FE). In order to show that t Ñ etA is continuous, we first observe that by (FE) one has et`h ´ etA “ etApehA ´ Iq for all t; h P R. Therefore, it hA suffices to show that limhÑ0e “ I: This follows from the estimate 8 hkAk }ehA ´ I}“} } ¤ k! k“1 ¸ |h|k}A}k k “ 18 “ e|h|}A} ´ 1: k! ¸ tA Obviously, the range of the function t Ñ T ptq :“ e in MnpCq is a commu- tative semigroup of matrices depending continuously on the parameter t P R`. In fact, this is a straightforward consequence of the following decisive property: The mapping t Ñ T ptq is a homomorphism from the additive semigroup pR`; `q into the multiplicative semigroup MnpCq. Keeping this in mind, we start to use the following terminology. Definition 1.6. We call etA; t ¥ 0 the (one-parameter) semigroup generated by the matrix A P MnpCq. The def- inition, the continuity, and the functional equation (FE) hold for any real and even complex t. Then the map T : t Ñ etA extends to a continuous (even analytic) homomorphism from the additive group pR; `q (or pC; `q) into the multiplicative group GLpn; Cq of all invertible, com- plex n ˆ n matrices. We call etA; t P R the (one-parameter) group generated by A. Example 1.7. (i) The (semi) group generated by a diagonal matrix A “ diagpa1; : : : anq is given by etA “ diagpeta1 ; : : : ; etan q: 10 Chapter 1.
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