Semigroups of Contractions

Abstract

A strongly continuous semigroup $\mathbb{T}$ is called a contraction semigroup if % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaacY % hatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaiab-nj8 % ujab-bW90naaBaaaleaacGaxOniDaaqabaGccaGG8bGaaiiFaGqaai % aa+bcacqWFKjcHcaGFXaGae8ha3hdaaa!4E6D! \[ ||\mathbb{T}_t || \leqslant 1 \] for all t ≥0. This chapter is a continuation of the previous one: we present basic facts about unbounded operators and strongly continuous semigroups on Hilbert spaces, but now the emphasis is on contraction semigroups and their generators, which are called m-dissipative operators. We also discuss other important classes of operators (self-adjoint, positive and skew-adjoint operators) that arise as generators or as ingredients of generators of contraction semigroups.We also investigate some classes of self-adjoint differential operators: Sturm-Liouville operators and the Dirichlet Laplacian on various domains in ℝⁿ.