Reed-Simon's Theorem X.70: strong continuity of a vector-valued function

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I'm struggling with understanding the hypothesis of Thm X.70 of Reed and Simon's Methods of mathematical physics. They define
$$ C(t,s) = H(t)H(s)^-1 - I, $$
and state:




Theorem X.70. Let $mathcalH$ be a Hilbert space and let $I$ be an open interval in $mathbbR$. For each $t in I$, let $H(t)$ be a self-adjoint operator on $mathcalH$ so that $0 in rho(H(t))$ and



  1. The $H(t)$ have common domain $mathcalD$.

  2. For each $varphi in mathcalH, (t - s)^-1 C(t, s)varphi$ is uniformly strongly continuous and uniformly bounded in $s$ and $t$ for $t neq s$ lying in any fixed compact subinterval of $I$.

  3. For each $varphi in mathcalH$, $C(t)varphi = lim_s nearrow t (t - s)^-1C(t, s)varphi$ exists uniformly for $t$ in each compact subinterval and $C(t)$ is bounded and strongly continuous in $t$.

Then for all $s leq t$ in any compact subinterval of $I$ and any $varphi in mathcalH$,
$$ U(t, s) varphi = lim_k to infty U_k(t, s)varphi $$
exists uniformly in $s$ and $t$. Further, if $varphi_s in mathcalD$, then $varphi(t) = U(t, s)varphi_s$ is in $mathcalD$ for all $t$ and satisfies
$$ ifracddt varphi(t) = H(t) varphi(t), qquad varphi(s) = varphi_s $$
and $Vertvarphi(t)Vert = Vert varphi_s Vert $ for all $t geq s$.




I won't copy the definition of $U_k(t,s)$ since my question is about hypothesis 2 and 3. Initially I thought that 2. means that the operator-valued function $(t-s)^-1C(t,s)$ needs to be strongly continuous on $t$ and $s$ instead of the vector-valued function $(t-s)^-1C(t,s) varphi$. That is, for every $varphi in mathcalH$, the mappings $s mapsto (t-s)^-1C(t,s) varphi$ and $t mapsto (t-s)^-1C(t,s) varphi$ from $mathbbR$ to $mathcalH$ are continuous.



However, I feel I must have misunderstood it. If $(t-s)^-1C(t,s)$ is strongly continuous in $s$ and $t$, shouldn't the mappings $t mapsto Vert (t-s)^-1C(t,s) varphi Vert$ and $s mapsto Vert (t-s)^-1C(t,s) varphi Vert$ be continuous? And if that's the case, then $Vert (t-s)^-1C(t,s) varphi Vert$ should be uniformly bounded in $s,t$ for $s,t$ in any compact subinterval, doesn't it?



If my interpretation is wrong, I don't know what strong continuity could mean for a function from $mathbbR$ to $mathcalH$; I've tried to find it on Reed-Simon's book and googled it with no success. Can any of you give me some hint or reference?




functional-analysis operator-theory hilbert-spaces




share|cite|improve this question



















  • Strong continuity (in $(s,t)$) is assumed at points with $t neq s$ so your argument for boundedness is not valid.
    – Kavi Rama Murthy
    Aug 3 at 11:48















up vote
1
down vote

favorite












I'm struggling with understanding the hypothesis of Thm X.70 of Reed and Simon's Methods of mathematical physics. They define
$$ C(t,s) = H(t)H(s)^-1 - I, $$
and state:




Theorem X.70. Let $mathcalH$ be a Hilbert space and let $I$ be an open interval in $mathbbR$. For each $t in I$, let $H(t)$ be a self-adjoint operator on $mathcalH$ so that $0 in rho(H(t))$ and



  1. The $H(t)$ have common domain $mathcalD$.

  2. For each $varphi in mathcalH, (t - s)^-1 C(t, s)varphi$ is uniformly strongly continuous and uniformly bounded in $s$ and $t$ for $t neq s$ lying in any fixed compact subinterval of $I$.

  3. For each $varphi in mathcalH$, $C(t)varphi = lim_s nearrow t (t - s)^-1C(t, s)varphi$ exists uniformly for $t$ in each compact subinterval and $C(t)$ is bounded and strongly continuous in $t$.

Then for all $s leq t$ in any compact subinterval of $I$ and any $varphi in mathcalH$,
$$ U(t, s) varphi = lim_k to infty U_k(t, s)varphi $$
exists uniformly in $s$ and $t$. Further, if $varphi_s in mathcalD$, then $varphi(t) = U(t, s)varphi_s$ is in $mathcalD$ for all $t$ and satisfies
$$ ifracddt varphi(t) = H(t) varphi(t), qquad varphi(s) = varphi_s $$
and $Vertvarphi(t)Vert = Vert varphi_s Vert $ for all $t geq s$.




I won't copy the definition of $U_k(t,s)$ since my question is about hypothesis 2 and 3. Initially I thought that 2. means that the operator-valued function $(t-s)^-1C(t,s)$ needs to be strongly continuous on $t$ and $s$ instead of the vector-valued function $(t-s)^-1C(t,s) varphi$. That is, for every $varphi in mathcalH$, the mappings $s mapsto (t-s)^-1C(t,s) varphi$ and $t mapsto (t-s)^-1C(t,s) varphi$ from $mathbbR$ to $mathcalH$ are continuous.



However, I feel I must have misunderstood it. If $(t-s)^-1C(t,s)$ is strongly continuous in $s$ and $t$, shouldn't the mappings $t mapsto Vert (t-s)^-1C(t,s) varphi Vert$ and $s mapsto Vert (t-s)^-1C(t,s) varphi Vert$ be continuous? And if that's the case, then $Vert (t-s)^-1C(t,s) varphi Vert$ should be uniformly bounded in $s,t$ for $s,t$ in any compact subinterval, doesn't it?



If my interpretation is wrong, I don't know what strong continuity could mean for a function from $mathbbR$ to $mathcalH$; I've tried to find it on Reed-Simon's book and googled it with no success. Can any of you give me some hint or reference?




functional-analysis operator-theory hilbert-spaces




share|cite|improve this question



















  • Strong continuity (in $(s,t)$) is assumed at points with $t neq s$ so your argument for boundedness is not valid.
    – Kavi Rama Murthy
    Aug 3 at 11:48













up vote
1
down vote

favorite









up vote
1
down vote

favorite











I'm struggling with understanding the hypothesis of Thm X.70 of Reed and Simon's Methods of mathematical physics. They define
$$ C(t,s) = H(t)H(s)^-1 - I, $$
and state:




Theorem X.70. Let $mathcalH$ be a Hilbert space and let $I$ be an open interval in $mathbbR$. For each $t in I$, let $H(t)$ be a self-adjoint operator on $mathcalH$ so that $0 in rho(H(t))$ and



  1. The $H(t)$ have common domain $mathcalD$.

  2. For each $varphi in mathcalH, (t - s)^-1 C(t, s)varphi$ is uniformly strongly continuous and uniformly bounded in $s$ and $t$ for $t neq s$ lying in any fixed compact subinterval of $I$.

  3. For each $varphi in mathcalH$, $C(t)varphi = lim_s nearrow t (t - s)^-1C(t, s)varphi$ exists uniformly for $t$ in each compact subinterval and $C(t)$ is bounded and strongly continuous in $t$.

Then for all $s leq t$ in any compact subinterval of $I$ and any $varphi in mathcalH$,
$$ U(t, s) varphi = lim_k to infty U_k(t, s)varphi $$
exists uniformly in $s$ and $t$. Further, if $varphi_s in mathcalD$, then $varphi(t) = U(t, s)varphi_s$ is in $mathcalD$ for all $t$ and satisfies
$$ ifracddt varphi(t) = H(t) varphi(t), qquad varphi(s) = varphi_s $$
and $Vertvarphi(t)Vert = Vert varphi_s Vert $ for all $t geq s$.




I won't copy the definition of $U_k(t,s)$ since my question is about hypothesis 2 and 3. Initially I thought that 2. means that the operator-valued function $(t-s)^-1C(t,s)$ needs to be strongly continuous on $t$ and $s$ instead of the vector-valued function $(t-s)^-1C(t,s) varphi$. That is, for every $varphi in mathcalH$, the mappings $s mapsto (t-s)^-1C(t,s) varphi$ and $t mapsto (t-s)^-1C(t,s) varphi$ from $mathbbR$ to $mathcalH$ are continuous.



However, I feel I must have misunderstood it. If $(t-s)^-1C(t,s)$ is strongly continuous in $s$ and $t$, shouldn't the mappings $t mapsto Vert (t-s)^-1C(t,s) varphi Vert$ and $s mapsto Vert (t-s)^-1C(t,s) varphi Vert$ be continuous? And if that's the case, then $Vert (t-s)^-1C(t,s) varphi Vert$ should be uniformly bounded in $s,t$ for $s,t$ in any compact subinterval, doesn't it?



If my interpretation is wrong, I don't know what strong continuity could mean for a function from $mathbbR$ to $mathcalH$; I've tried to find it on Reed-Simon's book and googled it with no success. Can any of you give me some hint or reference?




functional-analysis operator-theory hilbert-spaces




share|cite|improve this question











I'm struggling with understanding the hypothesis of Thm X.70 of Reed and Simon's Methods of mathematical physics. They define
$$ C(t,s) = H(t)H(s)^-1 - I, $$
and state:




Theorem X.70. Let $mathcalH$ be a Hilbert space and let $I$ be an open interval in $mathbbR$. For each $t in I$, let $H(t)$ be a self-adjoint operator on $mathcalH$ so that $0 in rho(H(t))$ and



  1. The $H(t)$ have common domain $mathcalD$.

  2. For each $varphi in mathcalH, (t - s)^-1 C(t, s)varphi$ is uniformly strongly continuous and uniformly bounded in $s$ and $t$ for $t neq s$ lying in any fixed compact subinterval of $I$.

  3. For each $varphi in mathcalH$, $C(t)varphi = lim_s nearrow t (t - s)^-1C(t, s)varphi$ exists uniformly for $t$ in each compact subinterval and $C(t)$ is bounded and strongly continuous in $t$.

Then for all $s leq t$ in any compact subinterval of $I$ and any $varphi in mathcalH$,
$$ U(t, s) varphi = lim_k to infty U_k(t, s)varphi $$
exists uniformly in $s$ and $t$. Further, if $varphi_s in mathcalD$, then $varphi(t) = U(t, s)varphi_s$ is in $mathcalD$ for all $t$ and satisfies
$$ ifracddt varphi(t) = H(t) varphi(t), qquad varphi(s) = varphi_s $$
and $Vertvarphi(t)Vert = Vert varphi_s Vert $ for all $t geq s$.




I won't copy the definition of $U_k(t,s)$ since my question is about hypothesis 2 and 3. Initially I thought that 2. means that the operator-valued function $(t-s)^-1C(t,s)$ needs to be strongly continuous on $t$ and $s$ instead of the vector-valued function $(t-s)^-1C(t,s) varphi$. That is, for every $varphi in mathcalH$, the mappings $s mapsto (t-s)^-1C(t,s) varphi$ and $t mapsto (t-s)^-1C(t,s) varphi$ from $mathbbR$ to $mathcalH$ are continuous.



However, I feel I must have misunderstood it. If $(t-s)^-1C(t,s)$ is strongly continuous in $s$ and $t$, shouldn't the mappings $t mapsto Vert (t-s)^-1C(t,s) varphi Vert$ and $s mapsto Vert (t-s)^-1C(t,s) varphi Vert$ be continuous? And if that's the case, then $Vert (t-s)^-1C(t,s) varphi Vert$ should be uniformly bounded in $s,t$ for $s,t$ in any compact subinterval, doesn't it?



If my interpretation is wrong, I don't know what strong continuity could mean for a function from $mathbbR$ to $mathcalH$; I've tried to find it on Reed-Simon's book and googled it with no success. Can any of you give me some hint or reference?





functional-analysis operator-theory hilbert-spaces





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Aug 3 at 11:24









Aitor B

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  • Strong continuity (in $(s,t)$) is assumed at points with $t neq s$ so your argument for boundedness is not valid.
    – Kavi Rama Murthy
    Aug 3 at 11:48

















  • Strong continuity (in $(s,t)$) is assumed at points with $t neq s$ so your argument for boundedness is not valid.
    – Kavi Rama Murthy
    Aug 3 at 11:48
















Strong continuity (in $(s,t)$) is assumed at points with $t neq s$ so your argument for boundedness is not valid.
– Kavi Rama Murthy
Aug 3 at 11:48





Strong continuity (in $(s,t)$) is assumed at points with $t neq s$ so your argument for boundedness is not valid.
– Kavi Rama Murthy
Aug 3 at 11:48
















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