Abstract
In this chapter we introduce the Fourier–Laplace transformation and use it to define operator-valued functions of ∂ t,ν; the so-called material law operators. These operators will play a crucial role when we deal with partial differential equations. In the equations of classical mathematical physics, like the heat equation, wave equation or Maxwell’s equation, the involved material parameters, such as heat conductivity or permeability of the underlying medium, are incorporated within these operators. Hence, these operators are also called “material law operators”. We start our chapter by defining the Fourier transformation and proving Plancherel’s theorem in the Hilbert space-valued case, which states that the Fourier transformation defines a unitary operator on \(L_2(\mathbb {R};H)\).
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In this chapter we introduce the Fourier–Laplace transformation and use it to define operator-valued functions of ∂ t,ν; the so-called material law operators. These operators will play a crucial role when we deal with partial differential equations. In the equations of classical mathematical physics, like the heat equation, wave equation or Maxwell’s equation, the involved material parameters, such as heat conductivity or permeability of the underlying medium, are incorporated within these operators. Hence, these operators are called “material law operators”. We start our chapter by defining the Fourier transformation and proving Plancherel’s theorem in the Hilbert space-valued case, which states that the Fourier transformation defines a unitary operator on \(L_2(\mathbb {R};H)\).
Throughout, let H be a complex Hilbert space.
5.1 The Fourier Transformation
We start by defining the Fourier transformation on \(L_{1}(\mathbb {R};H)\).
Definition
For \(f\in L_{1}(\mathbb {R};H)\) we define the Fourier transform \(\widehat {f}\) of f by
We also introduce
endowed with the sup-norm, \(\left \Vert \cdot \right \Vert { }_\infty \).
Lemma 5.1.1 ( Riemann–Lebesgue)
Let \(f\in L_{1}(\mathbb {R};H)\) . Then \(\widehat {f}\in C_{\mathrm {b}}(\mathbb {R};H)\) and \(\lim _{\left \vert t \right \vert \to \infty } \big \|\widehat {f}(t)\big \| = 0\) . Moreover,
Proof
First, note that \(\widehat {f}\) is continuous by dominated convergence and bounded with
This shows that the mapping
defines a bounded linear operator. Moreover, for \(\varphi \in C_{\mathrm {c}}^1(\mathbb {R};H)\) we compute
for s≠0 and thus,
which shows that \(\lim _{\left \vert s \right \vert \to \infty }\left \Vert \widehat {\varphi }(s) \right \Vert =0\). By the facts that \(C_{\mathrm {c}}^1(\mathbb {R};H)\) is dense in \(L_{1}(\mathbb {R};H)\) (see Lemma 3.1.8), \(\left \{ f\in C_{\mathrm {b}}(\mathbb {R};H) \,;\, \lim _{\left \vert t \right \vert \to \infty }\left \Vert f(t) \right \Vert =0 \right \}\) is a closed subspace of \(C_{\mathrm {b}}(\mathbb {R};H)\) and the operator in (5.1) is bounded, the assertion follows. □
It is our main goal to extend the definition of the Fourier transformation to functions in \(L_2(\mathbb {R};H)\). For doing so, we make use of the Schwartz space of rapidly decreasing functions.
Definition
We define
to be the Schwartz space of rapidly decreasing functions on \(\mathbb {R}\) with values in H.
As usual we abbreviate .
Remark 5.1.2
\(\mathcal {S}(\mathbb {R};H)\) is a Fréchet space with respect to the seminorms
Moreover, \(\mathcal {S}(\mathbb {R};H)\subseteq \bigcap _{p\in \left [1,\infty \right ]}L_{p}(\mathbb {R};H)\). Indeed, \(\mathcal {S}(\mathbb {R};H)\subseteq L_{\infty }(\mathbb {R};H)\) by definition, and for \( f \in \mathcal {S}(\mathbb {R};H)\) and \(1\leqslant p<\infty \) we have that
Proposition 5.1.3
For \(f\in \mathcal {S}(\mathbb {R};H)\) we have \(\widehat {f}\in \mathcal {S}(\mathbb {R};H)\) and the mapping
is bijective. Moreover, for \(f,g\in L_{1}(\mathbb {R};H)\) we have that
Additionally, if \(f,\widehat {f}\in L_{1}(\mathbb {R};H)\) then
Proof
Let \(f\in \mathcal {S}(\mathbb {R};H)\). By Exercise 5.1 we have
and
Using these formulas, one can show that \(\widehat {f}\in \mathcal {S}(\mathbb {R};H).\) Since the bijectivity of the Fourier transformation on \(\mathcal {S}(\mathbb {R};H)\) would follow from (5.3), it suffices to prove the formulas (5.2) and (5.3). Let \(f,g\in L_{1}(\mathbb {R};H).\) Then we compute using Proposition 3.1.6 and Fubini’s theorem
which yields (5.2). For proving formula (5.3), we consider the function γ defined by for \(t\in \mathbb {R}\). Clearly, \(\gamma \in \mathcal {S}(\mathbb {R})\). We claim that \(\widehat {\gamma }=\gamma \). Indeed, we observe that γ solves the initial value problem y′ + ty = 0 subject to y(0) = 1; if we can show that \(\widehat {\gamma }\) solves the same initial value problem, then their equality would follow from the uniqueness of the solution. First, we observe that \(\widehat {\gamma }(0)=\frac {1}{\sqrt {2\pi }}\int _{\mathbb {R}}\mathrm {e}^{-\frac {t^{2}}{2}}\,\mathrm {d} t=1.\) Second, we compute using the formulas (5.4) and (5.5) that
Altogether, we have shown that \(\widehat {\gamma }\) solves the same initial value problem as γ and hence, \(\widehat {\gamma }=\gamma \). Let now \(f\in L_{1}(\mathbb {R};H)\) with \(\widehat {f}\in L_{1}(\mathbb {R};H),\) a > 0 and x ∈ H. Then we compute using (5.2)
for each \(s\in \mathbb {R}\). Since this holds for all x ∈ H we get
Letting a → 0 in the latter equality, we obtain
where we have used dominated convergence for the term on the left-hand side. In order to compute the limit on the right-hand side, we first observe that
and hence, for each a > 0 the operator
is bounded by \(\left \Vert \gamma \right \Vert { }_{1}\). Moreover, since \(S_{a}\psi \to \psi (\cdot )\left \Vert \gamma \right \Vert { }_{1}\) as a → 0 for \(\psi \in C_{\mathrm {c}}(\mathbb {R};H)\), we infer that
for each \(f\in L_{1}(\mathbb {R};H)\). Hence, passing to a suitable sequence \(\left (a_{n}\right )_{n}\) in \(\mathbb {R}_{>0}\) tending to 0, we get
Using this identity for the right-hand side of (5.6), we get
and since \(\left \Vert \gamma \right \Vert { }_{1}=\sqrt {2\pi }\), we derive (5.3). □
With these preparations at hand, we are now able to prove the main theorem of this section.
Theorem 5.1.4 (Plancherel )
The mapping
extends to a unitary operator on \(L_2(\mathbb {R};H)\) , again denoted by \(\mathcal {F}\) , the Fourier transformation . Moreover, \(\mathcal {F}^{\ast }=\mathcal {F}^{-1}\) is given by \(f\mapsto \widehat {f}(-\cdot )\).
Proof
Using (5.2) and (5.3) we obtain that
for all \(f,g\in \mathcal {S}(\mathbb {R};H)\) and thus, in particular,
Moreover, \(\operatorname {dom}(\mathcal {F})=\operatorname {ran}(\mathcal {F})=\mathcal {S}(\mathbb {R};H)\) is dense in \(L_2(\mathbb {R};H)\) and hence, the first assertion follows by Exercise 5.2. As \(\mathcal {F}\) is unitary, we have \(\mathcal {F}^*=\mathcal {F}^{-1}\), thus, by (5.2) applied to \(f,g\in \mathcal {S}(\mathbb {R};H)\), we read off (using Proposition 2.3.8) that \(\mathcal {F}^{-1}=(f\mapsto \widehat {f}(-\cdot ))\), which yields all the claims of the theorem at hand. □
Remark 5.1.5
We emphasise that for \(f\in L_2(\mathbb {R};H)\) the Fourier transform \(\mathcal {F}f\) is not given by the integral expression for L 1-functions, simply because the integral does not need to exist. However, by dominated convergence
where the limit is taken in \(L_2(\mathbb {R};H).\)
5.2 The Fourier–Laplace Transformation and Its Relation to the Time Derivative
We now use the Fourier transformation to define an analogous transformation on our exponentially weighted L 2-type spaces; the so-called Fourier–Laplace transformation. We recall from Corollary 3.2.5 that for \(\nu \in \mathbb {R}\) the mapping
is unitary. In a similar fashion, we obtain that
defines an isometry.
Definition
Let \(\nu \in \mathbb {R}\). We define the Fourier–Laplace transformation as
We can also consider the Fourier–Laplace transformation as a mapping from \(L_{1,\nu }(\mathbb {R};H)\) to \(C_{\mathrm {b}}(\mathbb {R};H)\); that is,
Remark 5.2.1
Note that \(\mathcal {L}_{\nu }=\mathcal {F}\exp (-\nu \mathrm {m})\) is unitary as an operator from \(L_{2,\nu }(\mathbb {R};H)\) to \(L_2(\mathbb {R};H)\) since it is the composition of two unitary operators. For \(\varphi \in C_{\mathrm {c}}^\infty (\mathbb {R};H)\), we have the expression
which shows that \(\mathcal {L}_{\nu }\) can be interpreted as a shifted variant of the Fourier transformation, where the real part in the exponent equals ν instead of zero.
Our next goal is to show that the Fourier–Laplace transformation provides a spectral representation of our time derivative, ∂ t,ν.
Definition
Let \(V\colon \mathbb {R}\to \mathbb {K}\) be measurable. We define the multiplication-by-V operator as
with
In particular, if V is the identity on \(\mathbb {R}\) we will just write m instead of \(\operatorname {id}(\mathrm {m})\) and call it the multiplication-by-the-argument operator.
Remark 5.2.2
Note that the multiplication-by-V operator is a vector-valued analogue of the multiplication operator seen in Theorems 2.4.3 and 2.4.7. The statements in these theorems generalise (easily) to the vector-valued situation at hand. Thus, as in Theorem 2.4.3, one shows that m is selfadjoint. Moreover, when H ≠ {0}, in a similar fashion to the arguments carried out in Theorem 2.4.7 one shows that
In order to avoid trivial cases, we shall assume throughout that H ≠ {0}.
Theorem 5.2.3
Let \(\nu \in \mathbb {R}\) . Then
In particular,
Proof
We first prove the assertion for ν≠0 and show that
The assertion will then follow by Theorem 2.4.3(d). Note that \(\frac {1}{\mathrm {i}\mathrm {m}+\nu }\in L(L_2(\mathbb {R};H))\) by Proposition 2.4.6, and hence, both operators I ν and \(\mathcal {L}_{\nu }^{\ast }(\tfrac {1}{\mathrm {i}\mathrm {m}+\nu })\mathcal {L}_{\nu }\) are bounded and defined on the whole of \(L_{2,\nu }(\mathbb {R};H).\) Thus, it suffices to prove the equality on a dense subset of \(L_{2,\nu }(\mathbb {R};H)\), like \(C_{\mathrm {c}}(\mathbb {R};H).\) We will just do the computation for the case when ν > 0. So, let \(\varphi \in C_{\mathrm {c}}(\mathbb {R};H)\) and compute
for \(t\in \mathbb {R}.\) For ν < 0 the computation is analogous. In the case when ν = 0 we observe that
□
5.3 Material Law Operators
Using the multiplication operator representation of ∂ t,ν via the Fourier–Laplace transformation, we can assign a functional calculus to this operator. We will do this in the following and define operator-valued functions of ∂ t,ν. The class of functions used for this calculus are the so-called material laws. We begin by defining this function class.
Definition
A mapping \(M\colon \operatorname {dom}(M)\subseteq \mathbb {C}\to L(H)\) is called a material law if
-
(a)
\(\operatorname {dom}(M)\) is open and M is holomorphic (i.e., complex differentiable; see also Exercise 5.3),
-
(b)
there exists some \(\nu \in \mathbb {R}\) such that \(\mathbb {C}_{\operatorname {Re}>\nu }\subseteq \operatorname {dom}(M)\) and
Moreover, we set
to be the abscissa of boundedness of M.
Example 5.3.1
Let us state various examples of material laws.
-
(a)
Polynomials in z −1: Let \(n\in \mathbb {N}_0\), M 0, …, M n ∈ L(H). Then
defines a material law with
$$\displaystyle \begin{aligned} \mathrm{s}_{\mathrm{b}}\left( M \right) & =\begin{cases} -\infty & \text{ if }M_{1}=\ldots=M_{n}=0,\\ 0 & \text{ otherwise.} \end{cases} \end{aligned} $$ -
(b)
Series in z −1: Let \((M_{k})_{k\in \mathbb {N}}\) in L(H) such that \(\sum _{k=0}^{\infty }\left \Vert M_{k} \right \Vert r^{-k}<\infty \) for some r > 0. Then
defines a material law with \(\mathrm {s}_{\mathrm {b}}\left ( M \right )\leqslant r\).
-
(c)
Exponentials: Let \(h\in \mathbb {R},M_{0}\in L(H)\) where M 0≠0 and set
Then M is a material law if and only if h⩽0. In this case, \(\mathrm {s}_{\mathrm {b}}\left ( M \right )=-\infty .\)
-
(d)
Laplace transforms: Let \(\nu \in \mathbb {R}\) and \(k\in L_{1,\nu }(\mathbb {R})\) with \( \operatorname {\mathrm {spt}} k\subseteq \mathbb {R}_{\ge {0}}\). Then
defines a material law with \(\mathrm {s}_{\mathrm {b}}\left ( M \right )\leqslant \nu \).
-
(e)
Fractional powers: Let M 0 ∈ L(H), M 0≠0, \(\alpha \in \mathbb {R}\) and set
where we set
Then M is a material law if and only if α⩾0 and
$$\displaystyle \begin{aligned} \mathrm{s}_{\mathrm{b}}\left( M \right) & =\begin{cases} -\infty & \text{ if }\alpha=0,\\ 0 & \text{ otherwise}. \end{cases} \end{aligned} $$
For material laws M we now define the corresponding material law operators in terms of the functional calculus induced by the spectral representation of ∂ t,ν.
Proposition 5.3.2
Let \(M\colon \operatorname {dom}(M)\subseteq \mathbb {C}\to L(H)\) be a material law. Then, for \(\nu >\mathrm {s}_{\mathrm {b}}\left ( M \right )\) , the operator
is bounded. Moreover, we define the material law operator
and obtain
Proof
The proof is clear. □
Remark 5.3.3
The set of material laws is an algebra and the mapping of assigning a material law to its corresponding material law operator is an algebra homomorphism in the following sense. For j ∈{1, 2} let \(M_j\colon \operatorname {dom}(M_j)\subseteq \mathbb {C} \to L(H)\) be material laws, \(\lambda \in \mathbb {C}\). Then M 1 + M 2 (with domain \(\operatorname {dom}(M_1)\cap \operatorname {dom}(M_2)\)), λM 1 and M 1 ⋅ M 2 (with domain \(\operatorname {dom}(M_1)\cap \operatorname {dom}(M_2)\)) are material laws as well. Moreover, \(\mathrm {s}_{\mathrm {b}}\left ( M_1+M_2 \right ), \mathrm {s}_{\mathrm {b}}\left ( M_1\cdot M_2 \right )\leqslant \max \{\mathrm {s}_{\mathrm {b}}\left ( M_1 \right ),\mathrm {s}_{\mathrm {b}}\left ( M_2 \right )\}\). Furthermore, if M 2(z) is a scalar for all \(z\in \operatorname {dom}(M_2)\), then for \(\nu >\max \{\mathrm {s}_{\mathrm {b}}\left ( M_1 \right ),\mathrm {s}_{\mathrm {b}}\left ( M_2 \right )\}\) we have (M 1 M 2)(∂ t,ν) = M 1(∂ t,ν)M 2(∂ t,ν) = M 2(∂ t,ν)M 1(∂ t,ν) = (M 2 M 1)(∂ t,ν).
Example 5.3.4
We now revisit the material laws presented in Example 5.3.1 and compute their corresponding operators, M(∂ t,ν).
-
(a)
Let \(n\in \mathbb {N}_0\), M 0, …, M n ∈ L(H) and
Then, for ν > 0, one obviously has
$$\displaystyle \begin{aligned} M(\partial_{t,\nu})=\sum_{k=0}^{n}\partial_{t,\nu}^{-k}M_{k}, \end{aligned}$$due to Theorem 5.2.3.
-
(b)
Let \((M_{k})_{k\in \mathbb {N}}\) in L(H) such that \(\sum _{k=0}^{\infty }\left \Vert M_{k} \right \Vert r^{-k}<\infty \) for some r > 0 and
Then, for ν > r, one has
$$\displaystyle \begin{aligned} M(\partial_{t,\nu})=\sum_{k=0}^{\infty}\partial_{t,\nu}^{-k}M_{k} \end{aligned}$$again on account of Theorem 5.2.3.
-
(c)
Let h⩽0, M 0 ∈ L(H) and
Then, for \(\nu \in \mathbb {R}\), we have
$$\displaystyle \begin{aligned} M(\partial_{t,\nu})=M_{0}\tau_{h}, \end{aligned}$$where
$$\displaystyle \begin{aligned} \tau_{h}\colon L_{2,\nu}(\mathbb{R};H)\to L_{2,\nu}(\mathbb{R};H),\;f\mapsto\big(t\mapsto f(t+h)\big). \end{aligned}$$Indeed, for \(\varphi \in C_{\mathrm {c}}(\mathbb {R};H)\) we compute
$$\displaystyle \begin{aligned} \left(\mathcal{L}_{\nu}M_{0}\tau_{h}\varphi\right)(t) & =\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\mathrm{e}^{-(\mathrm{i} t+\nu)s}M_{0}\varphi(s+h)\,\mathrm{d} s \\ & =M_{0}\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\mathrm{e}^{-(\mathrm{i} t+\nu)\left(s-h\right)}\varphi(s)\,\mathrm{d} s =M(\mathrm{i} t+\nu)\left(\mathcal{L}_{\nu}\varphi\right)(t) \end{aligned} $$for all \(t\in \mathbb {R}\), where we have used Proposition 3.1.6 in the second line. Hence,
$$\displaystyle \begin{aligned} M_{0}\tau_{h}\varphi=\mathcal{L}_{\nu}^{\ast}M(\mathrm{i}\mathrm{m}+\nu)\mathcal{L}_{\nu}\varphi=M(\partial_{t,\nu})\varphi \end{aligned}$$and since \(C_{\mathrm {c}}(\mathbb {R};H)\) is dense in \(L_{2,\nu }(\mathbb {R};H)\) the assertion follows.
-
(d)
Let \(\nu \in \mathbb {R}\) and \(k\in L_{1,\nu }(\mathbb {R})\) with \( \operatorname {\mathrm {spt}} k\subseteq \mathbb {R}_{\ge {0}}\) and
Then, by Exercise 5.4,
$$\displaystyle \begin{aligned} M(\partial_{t,\mu})=k\ast \end{aligned}$$for each μ > ν.
-
(e)
Let M 0 ∈ L(H), α > 0 and
Then for ν > 0 we have
$$\displaystyle \begin{aligned} \left(M(\partial_{t,\nu})f\right)(t)=M_{0}\int_{-\infty}^{t}\frac{1}{\Gamma(\alpha)}(t-s)^{\alpha-1}f(s)\,\mathrm{d} s\quad (\text{a.e. }t\in\mathbb{R}){} \end{aligned} $$(5.8)for each \(f\in L_{2,\nu }(\mathbb {R};H)\); see Exercise 5.5. This formula gives rise to the definition
which is known as the (Riemann–Liouville) fractional integral of order α.
Throughout the previous examples, the operator M(∂ t,ν) did not depend on the actual value of ν. Indeed, this is true for all material laws. In order to see this, we need the following lemma.
Lemma 5.3.5
Let \(\mu ,\nu \in \mathbb {R}\) with μ < ν, and set . Let \(g\colon \overline {U}\to H\) be continuous and holomorphic on U such that \(g(\mathrm {i}\cdot +\nu ),g(\mathrm {i}\cdot +\mu )\in L_2(\mathbb {R};H)\) and there exists a sequence \((R_{n})_{n\in \mathbb {N}}\) in \(\mathbb {R}_{\geqslant 0}\) such that R n →∞ and
Then
Proof
Let \(t\in \mathbb {R}\). By Cauchy’s integral theorem, we have that
where \(\gamma _{R_{n}}\) is the rectangular closed path with corners ±iR n + μ, ±iR n + ν (see Fig. 5.1). Thus, we have that
Note that with the help of the formula for the inverse Fourier transformation (see Theorem 5.1.4) and \(\mathcal {L}_\nu ^*=(\mathcal {F}\exp (-\nu \mathrm {m}))^*=\exp (-\nu \mathrm {m} )^{-1}\mathcal {F}^*\) the left-hand side of (5.10) is nothing but
and hence, there is a subsequence of (R n)n (which we do not relabel) such that the left-hand side of (5.10) tends to
for almost every \(t\in \mathbb {R}\) as n →∞. As such, all we need to show is that the right-hand side of (5.10) tends to 0 as n →∞, which obviously follows by (5.9). □
Theorem 5.3.6
Let \(M\colon \operatorname {dom}(M)\subseteq \mathbb {C}\to L(H)\) be a material law. Then, for \(\mu ,\nu >\mathrm {s}_{\mathrm {b}}\left ( M \right )\) and \(f\in L_{2,\nu }(\mathbb {R};H)\cap L_{2,\mu }(\mathbb {R};H)\) , we have
Moreover, M(∂ t,ν) is causal for all \(\nu >\mathrm {s}_{\mathrm {b}}\left ( M \right )\).
Proof
Let μ < ν. We prove the assertion for with a < b and x ∈ H first. For \(\rho \in \mathbb {R}\) we compute
for all \(t\in \mathbb {R}\setminus \{0\}\). Moreover, we define
and prove that g satisfies the assumptions of Lemma 5.3.5. First, we note that g is bounded on \(\left \{ z\in \mathbb {C} \,;\, \mu \leqslant \operatorname {Re} z\leqslant \nu \right \}\setminus \{0\}\). Indeed, we only need to prove that it is bounded near 0 provided that μ⩽0. To that end, we observe
Thus, g is bounded near 0. In particular, z = 0 is a removable singularity and, hence, g can be extended holomorphically to \(\mathbb {C}_{\operatorname {Re}\geqslant \mu }\). Moreover, for \(\rho \geqslant \mu \) we have that
The first term on the right-hand side is finite since g is bounded, while the second term can be estimated by
This proves that \(g(\mathrm {i}\cdot +\rho )\in L_2(\mathbb {R};H)\) for each \(\rho \geqslant \mu \) and hence, particularly for ρ = μ and ρ = ν. Finally, for \(\rho \geqslant \mu \) we have that
which together with the boundedness of g yields (5.9) by dominated convergence. This shows that g satisfies the assumptions of Lemma 5.3.5 and thus
By linearity, this equality extends to \(S_{\mathrm {c}}(\mathbb {R};H)\) and so,
is well-defined. Moreover, F is uniformly Lipschitz continuous (observe that \(\sup _{\nu \geqslant \mu } \left \Vert F^{\nu } \right \Vert \leq \left \Vert M \right \Vert { }_{\infty ,\mathbb {C}_{\operatorname {Re}>\mu }}\)) and hence, the assertions follow from Lemma 4.2.5. □
5.4 Comments
The Fourier and the Fourier–Laplace transformation introduced in this chapter are used to define an operator-valued functional calculus for the time derivative, ∂ t,ν. This functional calculus can be defined since the Fourier–Laplace transformation provides the unitary transformation yielding the spectral representation of the time derivative as multiplication operator. This fact was already noticed in [83], which eventually led to evolutionary equations in [82].
We emphasise that we have used the fundamental property that both \(\mathcal {F}\) and \(\mathcal {L}_{\nu }\) are unitary. It is noteworthy that the Fourier transformation is an isometric isomorphism on \(L_{2}(\mathbb {R};X)\) if and only if X is a Hilbert space, see [58]. In the Banach space-valued case one has to further restrict the class of functions used to define a functional calculus. For the topic of functional calculus we refer to the 21st Internet Seminar [46] by Markus Haase and to his monograph, [47].
Material laws and the corresponding material law operators were also considered in [82, Section 3], including a physical motivation. Note that the definition in [82] is slightly different compared to the one presented here.
Exercises
Exercise 5.1
Let ( Ω, Σ, μ) be a σ-finite measure space, X a Banach space and \(I\subseteq \mathbb {R}\) an open interval. Let g: I × Ω→ X such that g(t, ⋅) ∈ L 1(μ;X) for each t ∈ I, and define
-
(a)
Assume that g(⋅, ω) is continuous for μ-almost every ω ∈ Ω and let f ∈ L 1(μ) such that
$$\displaystyle \begin{aligned} \left\Vert g(t,\omega) \right\Vert \leqslant f(\omega)\quad (t\in I,\omega\in\Omega). \end{aligned}$$Prove that h is continuous.
-
(b)
Assume that g(⋅, ω) is differentiable for μ-almost every ω ∈ Ω and let f ∈ L 1(μ) such that
$$\displaystyle \begin{aligned} \left\Vert \partial_{t}g(t,\omega) \right\Vert \leqslant f(\omega)\quad (t\in I,\mu-a.a.\,\,\, \omega\in\Omega). \end{aligned}$$Prove that h is differentiable with
$$\displaystyle \begin{aligned} h'(t)=\int_{\Omega}\partial_{t}g(t,\omega)\,\mathrm{d}\mu(\omega). \end{aligned}$$
Exercise 5.2
Let H 0, H 1 be two Hilbert spaces and \(U\colon \operatorname {dom}(U)\subseteq H_{0}\to H_{1}\) linear such that
-
\(\operatorname {dom}(U)\) is dense in H 0 and \(\operatorname {ran}(U)\) is dense in H 1.
-
\(\forall x\in \operatorname {dom}(U):\:\left \Vert Ux \right \Vert { }_{H_{1}}=\left \Vert x \right \Vert { }_{H_{0}}\).
Show that U can be uniquely extended to a unitary operator between H 0 and H 1.
Exercise 5.3
Let \(\Omega \subseteq \mathbb {C}\) be open, X a complex Banach space and f : Ω → X. Prove that the following statements are equivalent:
-
(i)
f is holomorphic.
-
(ii)
For all x′∈ X′ the mapping \(x'\circ f\colon \Omega \to \mathbb {C}\) is holomorphic.
-
(iii)
f is locally bounded and \(x'\circ f\colon \Omega \to \mathbb {C}\) is holomorphic for all x′∈ D, where D ⊆ X′ is a norming setFootnote 1 for X.
-
(iv)
f is analytic, i.e. for each z 0 ∈ Ω there is r > 0 and (a n)n in X with \(B\left (z_{0},r\right )\subseteq \Omega \) and
$$\displaystyle \begin{aligned} f(z)=\sum_{n=0}^{\infty}a_{n}\left(z-z_{0}\right)^{n}\quad (z\in B\left(z_{0},r\right)).\end{aligned} $$
Assume now that X = L(X 1, X 2) for two complex Banach spaces X 1, X 2, let D 1 ⊆ X 1 be dense and \(D_{2}\subseteq X_{2}^{\prime }\) norming for X 2. Prove that the statements (i) to (iv) are equivalent to
-
(v)
f is locally bounded and \(\Omega \ni z\mapsto x_{2}^{\prime }(f(z)(x_{1}))\in \mathbb {C}\) is holomorphic for all x 1 ∈ D 1 and \(x_2^{\prime }\in D_2\).
Hint: For the difficult implications one might also consult [6, Appendix A]. In the same source one can find that in part (iii) it is enough for D to be separating.
Exercise 5.4
Let \(\nu \in \mathbb {R}\) and \(k\in L_{1,\nu }(\mathbb {R})\). Prove that
for \(f\in L_{2,\nu }(\mathbb {R};H)\).
Exercise 5.5
Let α > 0 and define for \(t\in \mathbb {R}.\) Show that \(g_{\alpha }\in L_{1,\nu }(\mathbb {R})\) for each ν > 0 and that
Use this formula and Exercise 5.4 to derive (5.8).
Hint: To compute the Fourier–Laplace transform of g α, derive that \(\mathcal {L}_{\nu }g_{\alpha }\) solves a first order ordinary differential equation and use separation of variables to solve this equation.
Exercise 5.6
Let \(\mu ,\nu \in \mathbb {R}\) with μ < ν and \(f\in L_{2,\nu }(\mathbb {R};H)\cap L_{2,\mu }(\mathbb {R};H)\). Moreover, set . Show that \(f\in \bigcap _{\mu <\rho <\nu }L_{2,\rho }(\mathbb {R};H)\cap L_{1,\rho }(\mathbb {R};H)\) and that
is holomorphic.
Exercise 5.7
Let H 0, H 1 be Hilbert spaces and \(T\colon L_{2,\nu }(\mathbb {R};H_{0})\to L_{2,\nu }(\mathbb {R};H_{1})\) linear and bounded. We call T autonomous if Tτ h = τ h T for each \(h\in \mathbb {R}\) (τ h denotes the translation operator defined in Example 5.3.4). Prove that for autonomous T, the following statements are equivalent:
-
(i)
T is causal.
-
(ii)
For all \(f\in L_{2,\nu }(\mathbb {R};H_{0})\) with \( \operatorname {\mathrm {spt}} f\subseteq \left [0,\infty \right )\) one has \( \operatorname {\mathrm {spt}} Tf\subseteq \left [0,\infty \right )\).
Moreover, prove that for a material law M, the operator M(∂ t,ν) is autonomous for each \(\nu >\mathrm {s}_{\mathrm {b}}\left ( M \right )\).
Notes
- 1.
D ⊆ X′ is called a norming set for X if \(\left \Vert x \right \Vert =\sup _{x'\in D\setminus \{0\}}\frac {1}{\left \Vert x' \right \Vert }\left \vert x'(x) \right \vert \) for each x ∈ X. Note that X′ is norming for X by the Hahn–Banach theorem.
References
W. Arendt et al., Vector-Valued Laplace Transforms and Cauchy Problems, 2nd edn. (Birkhäuser, Basel, 2011)
M. Haase, Functional Calculus. 21st Internet Seminar. 2017/2018.
M. Haase, The Functional Calculus for Sectorial Operators, vol. 169 (Birkhäuser, Basel, 2006)
S. Kwapien, Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients. Stud. Math. 44(6), 583–595 (1972)
R. Picard, A structural observation for linear material laws in classical mathematical physics. Math. Method Appl. Sci. 32, 1768–1803 (2009)
R. Picard, Hilbert Space Approach to Some Classical Transforms (Wiley, New York, 1989)
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Seifert, C., Trostorff, S., Waurick, M. (2022). The Fourier–Laplace Transformation and Material Law Operators. In: Evolutionary Equations. Operator Theory: Advances and Applications, vol 287. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-89397-2_5
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