Summary
Section 6.1 is intended to give a characterization of the infinitely differentiable propagators of (ACP n ) in Banach spaces, which depends only on the properties of <m:math display='block'> <m:mrow> <m:msup> <m:mi>λ</m:mi> <m:mrow> <m:mi>k</m:mi><m:mo>−</m:mo><m:mn>1</m:mn></m:mrow> </m:msup> <m:mover accent='true'> <m:mrow> <m:msub> <m:mi>R</m:mi> <m:mi>λ</m:mi> </m:msub> <m:msub> <m:mi>A</m:mi> <m:mi>k</m:mi> </m:msub> </m:mrow> <m:mo stretchy='true'>¯</m:mo> </m:mover> <m:mrow><m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn><m:mo>≤</m:mo><m:mi>k</m:mi><m:mo>≤</m:mo><m:mi>n</m:mi></m:mrow> <m:mo>)</m:mo></m:mrow></m:mrow> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${\lambda ^{k - 1}}\overline {{R_\lambda }{A_k}} \left( {1 \leqslant k \leqslant n} \right)$$ . As a corollary, a concise sufficient condition is also presented.
Section 6.2 explores the characterization of the norm continuity (i.e., continuity in the uniform operator topology) for t > 0 of the propagators of (ACP n ) in Hilbert spaces. Following a general discussion on Laplace transforms in this respect, we obtain a succinct characterization (Theorem 2.1).
In Section 6.3, we restrict to (ACP 2) in a Banach space with A 1 ∈ L(E); see also Section 2.5. We show that S 0(t) or S 1 ′ (t) is norm continuous for t > 0 if and only if A 0 is bounded. This leads to an interesting consequence for strongly continuous cosine operator functions or operator groups.
Section 6.4 is concerned with the operator matrix <m:math display='block'> <m:mrow> <m:msub> <m:mi>A</m:mi> <m:mi>B</m:mi> </m:msub> <m:mo>=</m:mo><m:mrow><m:mo>(</m:mo> <m:mrow> <m:mtable> <m:mtr> <m:mtd> <m:mn>0</m:mn> </m:mtd> <m:mtd> <m:mrow> <m:msup> <m:mi>A</m:mi> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:mrow> </m:msup> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mrow> <m:mo>−</m:mo><m:msup> <m:mi>A</m:mi> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:mrow> </m:msup> </m:mrow> </m:mtd> <m:mtd> <m:mrow> <m:mo>−</m:mo><m:mi>B</m:mi></m:mrow> </m:mtd> </m:mtr> </m:mtable></m:mrow> <m:mo>)</m:mo></m:mrow><m:mo>,</m:mo></m:mrow> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${A_B} = \left( {\begin{array}{*{20}{c}} 0&{{A^{\frac{1}{2}}}} \\ { - {A^{\frac{1}{2}}}}&{ - B} \end{array}} \right),$$ , where A is a positive self-adjoint operator in a Hilbert space and B subordinated to A in various ways. One can see that the semigroup generated by A B (or <m:math display='block'> <m:mrow> <m:mover accent='true'> <m:mrow> <m:msub> <m:mi>A</m:mi> <m:mi>B</m:mi> </m:msub> </m:mrow> <m:mo stretchy='true'>¯</m:mo> </m:mover> </m:mrow> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\overline {{A_B}} $$ ) may possess norm continuity, differentiability, analyticity, or exponential stability, respectively, as B changes.