Let $T(t)$ be a $C_0$-semigroup on Banach space $X$, and $A$ its generator. By Lumer-Philipps theorem we know that if $A$ is densely defined and m-dessipative operator then it generates a $C_0$-semigroup of contractions, I.e., $$\|T(t)\| \leq 1, \quad \forall t \geq 0.$$ My question here: is there any theorem which gives conditions to generate a strictly contraction $C_0$-semigroup? That is, $$\|T(t)\| < 1, \quad \forall t > 0.$$ For example, for $A=\Delta$ the Dirichlet Laplacian, is the associated semigroup strictly contractive? Can we calculate the norm $\|T(t)\|$ in this case? Or at least, Is $I-T(t)$ invertible? If this is true for the Dirichlet laplacian, can we obtain the same result for second order elliptic operator?

Let $T(t)$ be a $C_0$-semigroup on Banach space $X$, and $A$ its generator. By Lumer-Philipps theorem we know that if $A$ is densely defined and m-dissipative operator then it generates a $C_0$-semigroup of contractions, i.e., $$\|T(t)\| \leq 1, \quad \forall t \geq 0.$$ My question here: is there any theorem which gives conditions to generate a strictly contraction $C_0$-semigroup? That is, $$\|T(t)\| < 1, \quad \forall t > 0.$$ For example, for $A=\Delta$ the Dirichlet Laplacian, is the associated semigroup strictly contractive? Can we calculate the norm $\|T(t)\|$ in this case? Or at least, Is $I-T(t)$ invertible? If this is true for the Dirichlet laplacian, can we obtain the same result for second order elliptic operator?

Let $T(t)$ be a $C_0$-semigroup on Banach space $X$, and $A$ its generator. By Lumer-Philipps theorem we know that if $A$ is densely defined and m-dessipative operator then it generates a $C_0$-semigroup of contractions, I.e., $$\|T(t)\| \leq 1, \quad \forall t \geq 0.$$ My question here: is there any theorem which gives conditions to generate a strictly contraction $C_0$-semigroup? That is, $$\|T(t)\| < 1, \quad \forall t \geq 0.$$ For example, for $A=\Delta$ the Dirichlet Laplacian, is the associated semigroup strictly contractive? Can we calculate the norm $\|T(t)\|$ in this case? Or at least, Is $I-T(t)$ invertible? If this is true for the Dirichlet laplacian, can we obtain the same result for second order elliptic operator?

Let $T(t)$ be a $C_0$-semigroup on Banach space $X$, and $A$ its generator. By Lumer-Philipps theorem we know that if $A$ is densely defined and m-dessipative operator then it generates a $C_0$-semigroup of contractions, I.e., $$\|T(t)\| \leq 1, \quad \forall t \geq 0.$$ My question here: is there any theorem which gives conditions to generate a strictly contraction $C_0$-semigroup? That is, $$\|T(t)\| < 1, \quad \forall t > 0.$$ For example, for $A=\Delta$ the Dirichlet Laplacian, is the associated semigroup strictly contractive? Can we calculate the norm $\|T(t)\|$ in this case? Or at least, Is $I-T(t)$ invertible? If this is true for the Dirichlet laplacian, can we obtain the same result for second order elliptic operator?

Let $T(t)$ be a $C_0$-semigroup on Banach space $X$, and $A$ its generator. By Lumer-Philipps theorem we know that if $A$ is densly defined and m-dessipative operator then it generates a $C_0$-semigroup of contractions, I.e., $$||T(t)|| \leq 1, \quad \forall t \geq 0.$$ My question here: is there any theorem which gives conditions to generate a strictly contraction $C_0$-semigroup  ? That is, $$||T(t)|| < 1, \quad \forall t \geq 0.$$ For example, for $A=\Delta$ the Dirichlet laplacian, is the associated semigroup strictly contractive  ? Can we calculate the norm $\|T(t)\|$ in this case  ?. Or at least, Is $I-T(t)$ invertible  ? If this is true for the Dirichlet laplacian, can we obtain the same result for second order elliptic operator  ?

Let $T(t)$ be a $C_0$-semigroup on Banach space $X$, and $A$ its generator. By Lumer-Philipps theorem we know that if $A$ is densely defined and m-dessipative operator then it generates a $C_0$-semigroup of contractions, I.e., $$\|T(t)\| \leq 1, \quad \forall t \geq 0.$$ My question here: is there any theorem which gives conditions to generate a strictly contraction $C_0$-semigroup? That is, $$\|T(t)\| < 1, \quad \forall t \geq 0.$$ For example, for $A=\Delta$ the Dirichlet Laplacian, is the associated semigroup strictly contractive? Can we calculate the norm $\|T(t)\|$ in this case? Or at least, Is $I-T(t)$ invertible? If this is true for the Dirichlet laplacian, can we obtain the same result for second order elliptic operator?