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### Suppose that $T$ is a normal operator on $V$. Show that $\|T(v+w)\|=10$.

Suppose $T$ is a normal operator on $V$. Suppose also that $v, w \in V$ satisfy the equations $$\| v \|= \| w \| =2, Tv = 3v, Tw = 4w.$$ Show that $\| T(w+v) \| = 10.$ I thought this problem would ...
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### Why do we need an orthonormal basis to represent the adjoint of the operator?

For any linear operator on a finite dimensional Inner Product Space, we can get orthonormal basis via Gram Schmidt Process. But what is the necessity of defining the adjoint of the operator using ...
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I'm trying to show that the adjoint transformation $T^*$ of the endomorphism $T$ on a finite dimensional, real inner product space has the same characteristic polynomial as $T$ in a coordinate free (...
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### Why is $\sqrt{T^*T}$ self-adjoint?

Let $T$ be a bounded linear operator over some Hilbert space $H$. Since $T^*T$ is a positive operator, it has a square root. Let $R=\sqrt{T^*T}$. Prove that $\forall u\in H, ||Ru||=||Tu||$. Here'...
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### An identity regarding linear operators and their adjoint between Hilbert spaces

I came upon a trivial-seeming claim that I can't prove myself: Let $H_1$, $H_2$ and $H_3$ be finite-dimensional Hilbert spaces, let $A:H_1\rightarrow H_2$ and $B,C:H_3\rightarrow H_1$ be linear ...
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### Find the inverse of the $n\times n$ matrix whose entries are given by $a_{ij} = \max (i,j)$

The actual question on the past papers is "Let $n\ge 1$ be an integer and consider the $n\times n$ matrix $A$ whose entries are given by $a_{ij} = \max(i,j)$ for all $1\le i,j\le n$. Show that $A$ is ...
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### If A is included in B, what can I say about their adjoint?

I'm studying the theory of linear operators, but even if I tried to bruteforce my way I can't really continue without understanding this (I believe simple) statement: Let A and B be linear operators ...
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For $T$ a densely defined linear, not necessarily bounded operator on a Hilbert space $\mathscr{H}$, and $T^{**}$ the adjoint of $T$'s adjoint, I read somewhere that $Ran(T)=Ran(T^{**})$. Is that true?...
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### To prove norm on adjoint Banach space?

Case: which reminds me a lot about Bourbaki–Alaoglu theorem i.e. a unit case on a ball because of the last inequality and something with adjoint Hahn-Banach. My proposal by thinking the case on a ...
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### Give an example of a non-self-adjoint operator on a Hilbert space $H$ whose range is $H$ and which is not invertible.

Give an example of a non-self-adjoint operator on a Hilbert space $H$ whose range is $H$ and which is not invertible. I cannot think of an example to save my life. Any solutions/hints are greatly ...
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### How to prove this is a self-adjoint operator?

I have this operator from $H^1_0$ to $H^1_0$ defined by: $$Au(t)=\int_0^1 G(t,s) f(s,u(s))\mathsf ds$$ where $$G(t,s)=\begin{cases} t(1-s), &t\leq s\\s(1-t), &s\leq t.\end{cases}$$ I want to ...
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### On the space $l_2$ we define an operator $T$ by $Tx=(x_1, {x_2\over2}, {x_3\over3}, . . . )$. Show that $T$ is bounded, and find its adjoint. [duplicate]

On the space $l_2$ we define an operator $T$ by $Tx=(x_1, {x_2\over2}, {x_3\over3}, . . . )$. Show that $T$ is bounded I know that $||T||\leq 1$, but I don't know how to show this. Any solutions or ...
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### Is this operator $A = \pmatrix{1&1\\0&1}$ self-adjoint?

Is this operator $$A = \pmatrix{1&1\\0&1}$$ self-adjoint? I think not, because $$\pmatrix{1&1\\0&1}^T\neq A$$ where $T$ is the transposition of the matrix. What do you all think?
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### Proving the adjoint nature of operators using Hermiticity

How can the fact that $\hat x$ and $\hat p$ are Hermitian be used to prove that $\hat x - \frac{i}{m \omega} \hat p$ and $\hat x + \frac{i}{m \omega} \hat p$ are adjoints of each other?
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### When is the restriction of a normal operator not normal?

I was proving the spectral theorem for normal operators on finite-dimensional complex vector spaces today during a test, when I arrived at the point in which If $T\in\operatorname{End}(V)$ is ...
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### Work out the adjoint of $T(x,y) = (y,-x)$

this seems like a simple question but I don't understand it. We define a transformation $T(x,y) = (y,-x)$. We want to work out what the adjoint is. I know the answer: $T^*(x,y) = (-y,x)$ but how? ...
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### Confirm my understanding of adjoints

adjoints seem REALLY important and useful so I don't want to move onto the next topic without really understanding them; I have too many a times moved on and been lost because I don't have the ...
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### Find the adjoint of this non-standard inner product space

I'm really blanking out (a lot of late nights these past 10 weeks). The point of the exercise I'm about to type up is to show that the adjoint structure may possibly change when the inner product ...
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### Prob. 2, Sec. 3.9 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: Inversion and adjointness

Let $H$ be a Hilbert space, and let $T \colon H \to H$ be a bijective bounded linear operator whose inverse is bounded. Then how to show that $(T^*)^{-1}$ exists and $$(T^*)^{-1} = (T^{-1})^*?$$ My ...
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### Why is it called *adjunction* formula?

Let $X$ be a complex manifold, $Y$ a sub-manifold, and $i \colon Y \to X$ the corresponding embedding. Then one can prove that the corresponding canonical bundles satisfy  \omega_X \big|_Y = \...
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### What is the adjoint of an inverse matrix? [duplicate]

What is the adjoint of an inverse matrix? Is $(T^{-1})^{*} = (T^{*})^{-1}$?
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### $T \in B(X,Y)$ is an isometry if and only if $T^*$ is an isometry

I would like to prove that $T \in \mathscr{B}(X,Y)$ is an isometry of $X$ onto $Y$ if and only if $T^*$ is an isometry of $Y^*$ onto $X^*$. I am not really sure what to do. I started the argument as ...
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### Product of compact, bounded and self adjoint operator.

$T \in B(H)$, and $T = S^2$ for some self adjoint operator $S \in B(H)$. I need to prove that T is compact if and only if S is compact. If S is compact, it is easy to show that T is compact since S ...
Suppose $X$ and $Y$ are Banach spaces and $T:X\to Y$ is a bounded linear operator. Show that $T$ is an isometric isomorphism if and only if its adjoint $T^*$ is also an isometric isomorphism. Given an ...