Strongly continuous Operator.

Question

I can not understand the notion "strongly continuous operator on a Banach space". Is it the continuity with respect to norm on a Banach space?

Can someone kindly explain this to me along with reference?

Answer 1

Is it the continuity with respect to norm on a Banach space?

As explained in the Kantorovitz's book (Topics in Operator Semigroups) through the pages 3-6, the answer is yes.

Here is a (adapted) citation:

In the general situation, an operator semigroup is a function $T (\cdot) : [0,\infty) \to B(X)$ (where $B(X)$ denotes the Banach algebra of all bounded linear operators on the given Banach space $X$), such that

• $T (s)T (t) = T (s + t)$ for $s,t ≥ 0$ (the semigroup identity) and
• $T (0)$ is the identity operator $I$.

The continuity at $0$ (or $C_0$) condition is $$\lim_{t\to0^+}T (t)x = x$$ for all $x \in X$ (limit in $X$ with respect to the norm). This is right continuity at zero in the strong operator topology (s.o.t.) on $B(X)$; in brief, “strong continuity at $0$.”

A $C_0$-semigroup (or semigroup strongly right-continuous at zero) is a semigroup of operators that satisfies the $C_0$-condition.

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ADDENDUM. Maybe you find what you are looking for in the Engel's books (One-Parameter Semigroups for Linear Evolution Equations or A Short Course on Operator Semigroups).

Here is an adapted citation (from the first one):

... ($T(t))_{t\geq 0}$ is a strongly continuous semigroup if the functional equation $$\left\{\begin{align} &T(t + s) = T(t)T(s),\quad\text{ for all} \quad t, s \geq 0,\\ &T(0) = I \end{align}\right.\tag{FE}$$ holds and the orbit maps $$\xi_x:t\mapsto\xi_x(t):=T(t)x\tag{SC}$$ are continuous from $\mathbb{R}_+$ into $X$ for every $x\in X$.

The property $(\text{SC})$ can also be expressed by saying that the map $$t\mapsto T(t)$$ is continuous from $\mathbb{R}_+$ into the space $B(X)$ of all bounded operators on $X$ endowed with the strong operator topology...

These strongly continuous semigroups are the main objects in this book, and we are going to show how rich a theory and how many applications arise from the interplay of the functional equation $(\text{FE})$ and the requirement of strong continuity $(\text{SC})$...