Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$.
20a. Show that there exists a unique algebra morphism $\Psi_a : \mathcal{U}_q \rightarrow \mathcal{L}(W_{\ell})$ such that $$\forall \phi \in \mathcal{U}_q, \quad \Psi_a(\phi) \circ P_a = P_a \circ \phi$$
20b. Show that $\phi \in \mathcal{U}_q$ is contained in the kernel of $\Psi_a$ if and only if the image of $\phi$ is in the subspace of $V$ spanned by the vectors $v_i - a^p v_r, i \in \mathbf{Z}$, where $i = p\ell + r$ is the Euclidean division of $i$ by $\ell$.

Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$, and $\Psi_a : \mathcal{U}_q \rightarrow \mathcal{L}(W_{\ell})$ the unique algebra morphism such that $\Psi_a(\phi) \circ P_a = P_a \circ \phi$ for all $\phi \in \mathcal{U}_q$. We study $\Psi_a(E)$ in this question.
21a. Determine $\Psi_a(E)(v_0)$.
21b. Deduce $\Psi_a(E^{\ell})$.
21c. Calculate the dimension of the vector subspace $\mathbf{C}[\Psi_a(E)]$.
21d. Calculate the eigenvectors of $\Psi_a(E)$.

Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$, and $\Psi_a : \mathcal{U}_q \rightarrow \mathcal{L}(W_{\ell})$ the unique algebra morphism such that $\Psi_a(\phi) \circ P_a = P_a \circ \phi$ for all $\phi \in \mathcal{U}_q$. Let $W$ be a non-zero subspace of $W_{\ell}$ stable under $\Psi_a(H)$.
22a. Show that $W$ contains at least one of the vectors $v_i$.
22b. What can be said if $W$ is moreover stable under $\Psi_a(E)$?

Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$, and $\Psi_a : \mathcal{U}_q \rightarrow \mathcal{L}(W_{\ell})$ the unique algebra morphism such that $\Psi_a(\phi) \circ P_a = P_a \circ \phi$ for all $\phi \in \mathcal{U}_q$. Give a necessary and sufficient condition on $R(\lambda(0), \mu(0), q)$ for the operator $\Psi_a(F)$ to be nilpotent.

We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. Determine $p(n)$ as a function of the unique integer $d \in \mathbb{N}$ such that $n$ can be written as $n = 2^d m$ with $m$ odd.

Let $n \geqslant 1$ be a natural number. We equip $\mathbb{R}^n$ with the usual inner product and the usual Euclidean norm defined, for all $X = (x_1, \ldots, x_n)$ and $Y = (y_1, \ldots, y_n)$ of $\mathbb{R}^n$, by $$(X \mid Y) = \sum_{i=1}^n x_i y_i \quad \text{and} \quad \|X\| = \sqrt{\sum_{k=1}^n x_k^2}$$ We study the existence of a bilinear map $B_n: (\mathbb{R}^n)^2 \rightarrow \mathbb{R}^n$ satisfying $$\forall X, Y \in \mathbb{R}^n, \quad \|B_n(X, Y)\| = \|X\| \times \|Y\|$$ Show the existence of such a bilinear map $B_n$ when $n$ is one of the integers $1, 2, 4$.
For $n = 2$ (respectively 4) one may consider the product of two complex numbers (respectively of two quaternions).

Let $n \geqslant 1$ be a natural number. We equip $\mathbb{R}^n$ with the usual inner product and the usual Euclidean norm defined, for all $X = (x_1, \ldots, x_n)$ and $Y = (y_1, \ldots, y_n)$ of $\mathbb{R}^n$, by $$(X \mid Y) = \sum_{i=1}^n x_i y_i \quad \text{and} \quad \|X\| = \sqrt{\sum_{k=1}^n x_k^2}$$ We study the existence of a bilinear map $B_n: (\mathbb{R}^n)^2 \rightarrow \mathbb{R}^n$ satisfying $$\forall X, Y \in \mathbb{R}^n, \quad \|B_n(X, Y)\| = \|X\| \times \|Y\|$$ Using question II.B.2 show, for $n = 8$, the existence of a bilinear map satisfying the above. We do not ask you to explicitly write down a bilinear map $B_8$, but only to prove its existence.

Prove that $\left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle$ is indeed a quadratic form on $\mathbb { K } ^ { n }$, where $\left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle$ denotes the quadratic form $q$ defined on $\mathbb { K } ^ { n }$ by the formula $$q \left( x _ { 1 } , \ldots , x _ { n } \right) = a _ { 1 } x _ { 1 } ^ { 2 } + \cdots + a _ { n } x _ { n } ^ { 2 }$$

Two special cases. Let $d > 0$. Let $g \in \mathcal { C } ^ { 0 } ( [ 0 , d ] )$ such that $g ( 0 ) \neq 0$.
(a) Show that $$\int _ { 0 } ^ { d } e ^ { - t x } g ( x ) d x \underset { t \rightarrow + \infty } { \sim } \frac { g ( 0 ) } { t }$$ Hint. For $t > 0$, one can construct a function $g _ { t }$ piecewise continuous on $[ 0 , + \infty [$, bounded, such that $$\int _ { 0 } ^ { d } e ^ { - t x } g ( x ) d x = \frac { 1 } { t } \int _ { 0 } ^ { + \infty } e ^ { - x } g _ { t } ( x ) d x$$ (b) Show similarly that $$\int _ { 0 } ^ { d } e ^ { - t x ^ { 2 } } g ( x ) d x \underset { t \rightarrow + \infty } { \sim } \frac { \sqrt { \pi } } { 2 } \frac { g ( 0 ) } { \sqrt { t } }$$ Hint. We recall the equality $\int _ { 0 } ^ { + \infty } e ^ { - x ^ { 2 } } d x = \frac { \sqrt { \pi } } { 2 }$.

Let $E$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ equipped with the norm $\|.\|_{\infty}$: $$\|f\|_{\infty} = \max_{x \in [0,1]} |f(x)|$$ We denote by $T$ the application defined on $E$ such that: $$\forall f \in E, \quad \forall x \in [0,1], \quad T(f)(x) = x f\left(\frac{x}{2}\right)$$ Show that $T \in \mathcal{L}(E)$.

Let $E$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ equipped with the norm $\|.\|_{\infty}$: $$\|f\|_{\infty} = \max_{x \in [0,1]} |f(x)|$$ We denote by $T$ the application defined on $E$ such that: $$\forall f \in E, \quad \forall x \in [0,1], \quad T(f)(x) = x f\left(\frac{x}{2}\right)$$ Calculate the minimal possible value for the constant $M$ in the relation $\exists M \geq 0, \forall f \in E, \|T(f)\|_E \leq M\|f\|_E$.

Let $E$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ equipped with the norm $\|.\|_{\infty}$: $$\|f\|_{\infty} = \max_{x \in [0,1]} |f(x)|$$ We denote by $T$ the application defined on $E$ such that: $$\forall f \in E, \quad \forall x \in [0,1], \quad T(f)(x) = x f\left(\frac{x}{2}\right)$$ Determine $\operatorname{Ker}(T)$ and $\operatorname{Im}(T)$.

Let $E$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ equipped with the norm $\|.\|_2$: $$\|f\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$$ We denote by $T$ the application defined on $E$ such that: $$\forall f \in E, \quad \forall x \in [0,1], \quad T(f)(x) = x f\left(\frac{x}{2}\right)$$ Show that $T \in \mathcal{L}(E)$ with this norm.

Let $E$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ equipped with the norm $\|.\|_2$: $$\|f\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$$ We denote by $T$ the application defined on $E$ such that: $$\forall f \in E, \quad \forall x \in [0,1], \quad T(f)(x) = x f\left(\frac{x}{2}\right)$$ Calculate the minimal possible value for the constant $M$ in the relation $\exists M \geq 0, \forall f \in E, \|T(f)\|_2 \leq M\|f\|_2$. For this, you may consider the family $(f_n)_{n \geq 2}$ of elements of $E$ such that: (i) $f_n$ is piecewise affine, (ii) $f_n(0) = f_n\left(\frac{1}{2} - \frac{1}{n}\right) = f_n\left(\frac{1}{2} + \frac{1}{n^2}\right) = f_n(1) = 0$ and $f_n\left(\frac{1}{2}\right) = 1$.

Prove that the map $q \mapsto \widetilde { q }$ is a bijection from the set of quadratic forms on $V$ to the set of symmetric bilinear forms on $V$, where $\widetilde { q } : V \times V \rightarrow \mathbb { K }$ is defined by $( x , y ) \mapsto \widetilde { q } ( x , y ) = \frac { 1 } { 2 } ( q ( x + y ) - q ( x ) - q ( y ) )$.

Let $f \in \mathcal { C } ^ { 0 } ( [ a , b ] )$ such that $f ( a ) \neq 0$ and $\varphi \in \mathcal { C } ^ { 1 } ( [ a , b ] )$. For every parameter $t \in \mathbb { R }$, we denote $$F ( t ) = \int _ { a } ^ { b } e ^ { - t \varphi ( x ) } f ( x ) d x$$ Case where the phase $\varphi$ has no critical point in $[ a , b ]$. We assume that $\varphi ^ { \prime } ( x ) > 0$ for all $x \in [ a , b ]$.
(a) Show that $\Phi : x \mapsto \varphi ( x ) - \varphi ( a )$ is a bijection from $[ a , b ]$ onto an interval of the form $[ 0 , \beta ]$, and that it is of class $\mathcal { C } ^ { 1 }$.
(b) Show that $$F ( t ) \underset { t \rightarrow + \infty } { \sim } \frac { e ^ { - t \varphi ( a ) } f ( a ) } { \varphi ^ { \prime } ( a ) t }$$ Hint. One can reduce to the case treated in question 1a) using a change of variable.

Let $H = l^2(\mathbb{N})$, the vector space of real sequences that are square-summable: $$l^2(\mathbb{N}) = \left\{(u_n)_{n \in \mathbb{N}} \in \mathbb{R}^{\mathbb{N}}, \quad \sum_{n=0}^{+\infty} |u_n|^2 < +\infty\right\}$$ equipped with the norm: $$\|u\|_2 = \sqrt{\sum_{n=0}^{+\infty} |u_n|^2}$$ We denote by $S$, respectively $V$, the left shift application: $(Su)_n = u_{n-1}$ if $n \geq 1$ and $(Su)_0 = 0$, respectively the right shift: $(Vu)_n = u_{n+1}$ if $n \geq 0$ in $H = l^2(\mathbb{N})$.
Show that $S$ and $V$ belong to $\mathcal{L}(H)$.

Let $H = l^2(\mathbb{N})$, the vector space of real sequences that are square-summable: $$l^2(\mathbb{N}) = \left\{(u_n)_{n \in \mathbb{N}} \in \mathbb{R}^{\mathbb{N}}, \quad \sum_{n=0}^{+\infty} |u_n|^2 < +\infty\right\}$$ equipped with the norm: $$\|u\|_2 = \sqrt{\sum_{n=0}^{+\infty} |u_n|^2}$$ We denote by $S$, respectively $V$, the left shift application: $(Su)_n = u_{n-1}$ if $n \geq 1$ and $(Su)_0 = 0$, respectively the right shift: $(Vu)_n = u_{n+1}$ if $n \geq 0$ in $H = l^2(\mathbb{N})$.
Calculate the point spectrum of $S$ and $V$.

We now work in the space of bounded real sequences $F = l^{\infty}(\mathbb{N})$ equipped with the norm $$\|u\|_{\infty} = \sup_{n \in \mathbb{N}} |u_n|$$ We denote by $S$, respectively $V$, the left shift application: $(Su)_n = u_{n-1}$ if $n \geq 1$ and $(Su)_0 = 0$, respectively the right shift: $(Vu)_n = u_{n+1}$ if $n \geq 0$.
Show that $S$ and $V$ belong to $\mathcal{L}(F)$.

We now work in the space of bounded real sequences $F = l^{\infty}(\mathbb{N})$ equipped with the norm $$\|u\|_{\infty} = \sup_{n \in \mathbb{N}} |u_n|$$ We denote by $S$, respectively $V$, the left shift application: $(Su)_n = u_{n-1}$ if $n \geq 1$ and $(Su)_0 = 0$, respectively the right shift: $(Vu)_n = u_{n+1}$ if $n \geq 0$.
Calculate the point spectrum of $S$ and $V$ in $F$.

We now work in the space of bounded real sequences $F = l^{\infty}(\mathbb{N})$ equipped with the norm $$\|u\|_{\infty} = \sup_{n \in \mathbb{N}} |u_n|$$ We denote by $S$, respectively $V$, the left shift application: $(Su)_n = u_{n-1}$ if $n \geq 1$ and $(Su)_0 = 0$, respectively the right shift: $(Vu)_n = u_{n+1}$ if $n \geq 0$.
Calculate the spectrum of $S$ and $V$ in $F$.

Let $\mathcal { B } : = \left( e _ { 1 } , \ldots , e _ { n } \right)$ be a basis of $V$. We associate to every symmetric bilinear form $b$ on $V$ a symmetric matrix $\Phi _ { \mathcal { B } } ( b ) : = \left( b \left( e _ { i } , e _ { j } \right) \right) _ { i , j = 1 \ldots n }$ called the matrix of $b$ in the basis $\mathcal { B }$. We recall that $b \mapsto \Phi _ { \mathcal { B } } ( b )$ is an isomorphism between the vector space of symmetric bilinear forms on $V$ and that of square symmetric matrices of size $n$.
(a) Prove that a quadratic form $q$ on $V$ is non-degenerate if and only if the determinant $\operatorname { det } \left( \Phi _ { \mathcal { B } } ( \tilde { q } ) \right)$ is non-zero.
(b) What is the matrix of $\left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle$ in the canonical basis of $\mathbb { K } ^ { n }$ ? Deduce that $\left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle \in \mathcal { Q } \left( \mathbb { K } ^ { n } \right)$.

Let $f \in \mathcal { C } ^ { 0 } ( [ a , b ] )$ such that $f ( a ) \neq 0$ and $\varphi \in \mathcal { C } ^ { 1 } ( [ a , b ] )$. For every parameter $t \in \mathbb { R }$, we denote $$F ( t ) = \int _ { a } ^ { b } e ^ { - t \varphi ( x ) } f ( x ) d x$$ Case where the phase $\varphi$ has a critical point at $a$. We now assume that $\varphi \in \mathcal { C } ^ { 2 } ( [ a , b ] )$, $\varphi ^ { \prime } ( a ) = 0 , \varphi ^ { \prime \prime } ( a ) > 0$, and $\varphi ^ { \prime } ( x ) > 0$ for all $\left. \left. x \in \right] a , b \right]$.
(a) Show that the formula $\psi ( x ) = \sqrt { \varphi ( x ) - \varphi ( a ) }$ defines a function of class $\mathcal { C } ^ { 1 }$ on $[ a , b ]$. Calculate $\psi ^ { \prime } ( a )$.
(b) Show that $\psi$ is a bijection from $[ a , b ]$ onto an interval of the form $[ 0 , \beta ]$.
(c) Show that $$F ( t ) \underset { t \rightarrow + \infty } { \sim } \sqrt { \frac { \pi } { 2 \varphi ^ { \prime \prime } ( a ) } } \frac { e ^ { - t \varphi ( a ) } f ( a ) } { \sqrt { t } } .$$ Hint. One can reduce to the case treated in question 1b) using a change of variable.

We denote by $K$ the function defined from $[0,1]^2$ to $\mathbb{R}$ by the following relation: $K(s,t) = (1-s)t$ if $0 \leq t \leq s \leq 1$ and $K(s,t) = (1-t)s$ otherwise. We denote by $T$ the application defined on $E = C([0,1], \mathbb{R})$, equipped with the norm $\|.\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$, by the relation: $$\forall f \in E, \quad \forall s \in [0,1], \quad T(f)(s) = \int_0^1 K(s,t) f(t) \, dt$$ Show that $T \in \mathcal{L}(E)$.

We denote by $K$ the function defined from $[0,1]^2$ to $\mathbb{R}$ by the following relation: $K(s,t) = (1-s)t$ if $0 \leq t \leq s \leq 1$ and $K(s,t) = (1-t)s$ otherwise. We denote by $T$ the application defined on $E = C([0,1], \mathbb{R})$, equipped with the norm $\|.\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$, by the relation: $$\forall f \in E, \quad \forall s \in [0,1], \quad T(f)(s) = \int_0^1 K(s,t) f(t) \, dt$$ Let $f \in E$. By decomposing $T(f)$ into two integrals, show that $T(f)$ is a $C^2$ function and express $(T(f))'$ then $(T(f))''$.