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Let $L$ be some linear and compact operator on some Hilbert space. And denote by $L^*$ the adjoint operator. I need to bound for some $c>0$: $$\left\|(L^*L+c I)^{-\frac{1}{2}}L^*\right\|\leq1$$ ...
• 35
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### Why a matrix has to be invertible for $\operatorname{adj} A^T=(\operatorname{adj}A)^T$ to be true?

I read a theorem If $A$ is an invertible square matrix, then $\operatorname{adj} A^T= (\operatorname{adj} A)^T$. But after attempting to prove it myself and also reading the proof I am unable to ...
44 views

### Prove the following: If $T \in \mathcal{L}(V)$, and $v$ is an eigenvector of $T$, then $\overline{v}$ is an eigenvector of $T^*$.

I feel like this should be obvious, but I'm kind of stuck on how to prove it. I know that $v \in \text{null}(T - \lambda I)$, and that $\overline{\lambda}$ is an eigenvalue of $T^*$, but I don't ...
• 57
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### Show that for any $S$, if $ST=TS$, then $ST^*=T^*S$.

Let $V$ be a complex inner product space with $\dim(V)<\infty$. Let $T$ be a normal operator ($TT^*=T^*T$). Show that $V$ is the direct sum of $\text{null}(T)$ and $\text{range}(T)$. Show that for ...
• 934
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### Show that the set is the convex hull of the spectrum of $T$.

Let $V$ be a complex inner product space with $\dim V<\infty$. Let $T$ be a normal operator $(TT^*=T^*T)$. Show that the set of $\{\langle Tv, v\rangle: v\in V, \|v\|=1\}$ is the convex hull of the ...
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• 498
1 vote
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### Eigenvalues of an operator and its adjoint

I would like to verify the proof given in an answer #2 form here. The claim to prove was that $\lambda$ is an eigenvalue of $T$ if and only if $\overline{\lambda}$ is eigenvalue of the adjoint ...
• 103
1 vote
39 views

### Find self-adjoint of $P=|a \rangle \langle b |$

Let $P$ be an operator s.a. $P=|a \rangle \langle b |$ and $P|f \rangle = \langle b | f \rangle | a \rangle$. Find the self-adjoint and the $P^2$ operator. My attempt: We know to find the self ...
• 407
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### Question about operators from $\mathbb{R}^\mathbb{N}$ to/from a real separable Hilbert space.

Let $X \equiv \mathbb{R}^\mathbb{N}$ denote the space of real sequences with the product topology. Suppose that $T \colon \mathcal{H} \to X$ is a linear operator, where $\mathcal{H}$ is a real, ...
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• 1
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### Prove that $\{T_v: \|v\|\leq 1\}$ is a family of pointwise uniformly bounded functionals

Let $V,W$ be Hilbert spaces and $A:V\rightarrow W$, $B:W^*\rightarrow V^*$ linear operators such that $l(Av)=(Bl)(v)$ for all $v\in V$ and for all $l\in W^*$. For $v\in V$ define $T_v(l):=l(A(v))$ for ...
• 1,772
217 views

### Derivative of adjoint operator-valued function

Consider an infinite dimensional complex Hilbert space $H$. I think that for a bounded operator-valued function $A: x\mapsto A(x) \in \mathcal B(H)$, where $x\in \mathbb R$, we can define the ...
• 272
21 views

### Showing that $\Vert T^*T \Vert = \Vert T \Vert^2$ for $T$ a linear and bounded operator from a Hilbert space to itself [duplicate]

I am reading a book in which it is mentioned that for a bounded and linear operator $T: H \to H$, with $H$ a Hilbert space we have $\Vert T^{*}T\Vert = \Vert T \Vert^2$. The proof is left as an ...
59 views

1 vote
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### How can I finish my proof that $||A^*A||=||A||^2$?

Let $A:X\rightarrow X$ be a bounded linear operator on a Hilbert space $X$. I want to show that $||A^*A||=||A||^2$. My idea was the following: Let me consider $x\in X$ such that $||x||\leq 1$, then ...
• 1,772
50 views

### Spectral Representation of $T$

the spectral family of the operator $T:\ell^2\rightarrow\ell^2$ defined by $$T(\xi_1,\xi_2,\xi_3,....)=(\xi_1/1,\xi_2/2,\xi_3/3,....)$$ I am trying to get the application of Spectral theorem of ...
• 3,439
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### Bessel sequence ,frame sequence in Hilbert Space

This question was asked in my assignment of Functional Analysis and I am not able to make progress in few parts. Question: Let H be an Hilbert space and let $(f_i)_{i\in I}$ be a (finite or infinite) ...
39 views

### Different answers for adjoint of $T : \ell^2 \to \ell^2$ such that $T(x_1, x_2, \dots) = (c_1 x_1, c_2 x_2, \dots),$ where's the mistake?

$c = \{c_n\} \in \ell^{\infty}$ and $T : \ell^2 \to \ell^2$ is defined as $T(\{x_n\}) = \{c_n x_n\},$ all spaces are over $\mathbb{C}.$ It's easy to prove that $||T|| = ||c||,$ but I'm getting 2 ...
• 4,815
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### Can someone explain why we define adjoint?

Recently I've been reviewing linear algebra. The definition of adjoint of a linear map on an inner product space seems not really natrual. Looks like people use this to define normal and self-adjoint ...
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• 151
1 vote
21 views

### Finding adjoint of a Matrix equation

For $\alpha_i \in \mathbb{C}$, write down the adjoint of \alpha_1\left|V_1\right\rangle=\alpha_2 \Theta \Pi\left|V_2\right\rangle\left\langle V_3 \mid V_4\right\rangle+\alpha_3^*\left|V_5\right\...
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Let $\mathcal H$ be a Hilbert space and $A: \mathcal D_A \to \mathcal H$ be an unbounded linear operator. Suppose also that $\mathcal D_A$ is dense in $\mathcal H$. We define the adjoint of $A$, $A^*$ ...