For questions about adjoints, in the category-theoretic or inner-product-space sense, as well as about adjugate matrices.

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1answer
30 views

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.

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 ...
4
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3answers
77 views

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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1answer
12 views

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?
3
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2answers
32 views

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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1answer
23 views

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? ...
2
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1answer
52 views

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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3answers
22 views

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 ...
0
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1answer
26 views

Adjoint of $T_A = Ax$

Is it true that if $T_A(x) = Ax$ then $T^*_A(x) = A^*x$? I tried to prove this for the standard inner product $$ \newcommand{\innp}[2]{\left\langle #1,#2 \right\rangle} \innp{Ax}{x} = ...
0
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1answer
34 views

Proving facts about adjoints

Let $F$ denote $\mathbb R$ or $\mathbb C$. Let $T : V → W$, $S : V → W$ and let $R: U → V$ be linear transformations between inner product spaces $U$, $V$, $W$ over $F$. Verify the following facts: ...
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2answers
36 views

Prob. 8, Sec. 3.10 in Kreyszig's functional analysis book: An isometric linear operator has its adjoint as its left inverse

Let $H$ be a Hilbert space, and let $T \colon H \to H$ satisfy $$\langle Tx, Tx \rangle = \langle x, x \rangle \ \mbox{ for all } \ x \in H.$$ Then $T$ is bounded and norm $\Vert T \Vert = 1$ (unless ...
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1answer
20 views

Prob. 6, Sec. 3.10 in Kreyszig's functional analysis book: Powers of self-adjoint operators

Let $H$ be a Hilbert space. If $T \colon H \to H$ is a bounded self-adjoint linear operator and $T \neq 0$, then $T^n \neq 0$ for all $n \in \mathbb{N}$. How to show this? I've managed to show ...
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1answer
55 views

Prob. 10, Sec. 3.9 in Kreyszig's functional analysis book: The null space and adjoint of the right-shift operator

Let $(e_n)$ be a total orthonormal sequence in a separable Hilbert space $H$, let $T \colon H \to H$ be defined as follows: Since span of $(e_n)$ is dense in $H$, for every $x \in H$, we have $$x = ...
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0answers
26 views

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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0answers
24 views

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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1answer
89 views

What is the class of topological spaces $X$ such that the functors $\times X:\mathbf{Top}\to\mathbf{Top}$ have right adjoints?

For any topological space $X$, define a functor $\times X:\mathbf{Top}\to\mathbf{Top}$ by $Y\mapsto Y\times X$ (and acting on the hom-sets in the natural way). I know that if $X$ is locally compact, ...
2
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2answers
68 views

Right adjoint of covariant hom functor

I've constructed the left adjoint of the functor $\mathbf{Hom(A, -)}: \mathbf{Sets} \to \mathbf{Sets}$. Now I'm trying to prove that the functor does not have a right adjoint, but I'm not sure how to ...
0
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1answer
22 views

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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0answers
42 views

Do combinatorial species have adjoints?

A combinatorial species is a functor $F$ from the category $\mathbb{B}$ of finite sets and bijections to itself. What (if anything) can be said about adjunctions of species?
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2answers
42 views

Show $ \langle Tx,x \rangle \in \mathbb R$ for all $x \in H$ implies $T$ is self-adjoint

Show that a linear operator $T: H \rightarrow H$ is self adjoint if and only if $\langle Tx, x \rangle \in \mathbb R$ for all $x \in H$. You may use that the equality that for all $x,y \in H$ ...
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2answers
29 views

proof of all integer entries in an inverse matrix

"Prove that if $det(A)=1$ and all the entries in $A$ are integers, then all the entries in $A^{-1}$ are integers." I began by setting up the adjoint method for finding the inverse. $A^{-1} = \cfrac ...
12
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2answers
161 views

Does the forgetful functor from $\mathbf{TopGrp}$ to $\mathbf{Top}$ admit a left adjoint?

Let TopGrp be the category of topological groups (not necessarily $T_0$) and Top the category of topological spaces. Does the forgetful functor $U:\mathbf{TopGrp}\to\mathbf{Top}$ admit a left ...
3
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1answer
55 views

$T \in B(X,Y)$ 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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2answers
42 views

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 ...
2
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1answer
52 views

Adjoint operator on Banach space

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 ...
5
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1answer
99 views

$A^{-1}$ has integer entries if and only if the ${\rm det}\ (A) =\pm 1$

So, $A$ is a nxn matrix with integer entried. The question is to prove that $A^{-1}$ has all integer entries if and only if ${\rm det}\ (A) =\pm 1$ I know that $A^{-1}= {\rm adj}(A)/{\rm det}(A)$ ...
0
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1answer
23 views

About the self-adjoint extension of an operator.

Let $B$ be a selfadjoint extension of an operator $A$ on a Hilbert space $H$. Let $\varphi \in \ker(A^\ast-z_0)$. Then i want to show that $\varphi + (z- z_0)(B-z)^{-1} \varphi \in \ker(A^\ast-z)$. I ...
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0answers
38 views

Adjoint of a 3x4 matrix

How do I find the adjoint of this matrix? I am familiar with finding the adjoint of an $n x n$ matrix, but this has thrown me. $$A= \left( \begin{array}{ccc} 1&-1&0\\ 0&0&1\\ ...
3
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1answer
68 views

(Hopefully) Simple question about the exterior algebra functor

I have some (hopefully super) basic questions about the exterior algebra functor $$ \wedge:R\text{-Mod}\rightarrow R\text{-Alg}. $$ As I (think I) understand it, if one considers it as a functor ...
0
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1answer
13 views

Relation between a function and its norm

While reading up on Sturm-Liouville system theory, I came across something I didn't fully understand. At one point, in the midst of proving the existence of solutions to the Sturm-Liouvill problem, ...
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0answers
110 views

Adjoint of Exponential Map

If $\exp: T_p(G) \rightarrow G$ is the expoenential map of a lie group, then what does the adjoint operator (as in $\langle Ax,y\rangle=\langle x,A^*,y\rangle$) of the derivative of exp look like? ...
4
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2answers
85 views

Adjoint functor to an R-algebra only “remembering” itself as a ring

I have been wondering this question while trying to comprehend adjoint functors and the various definitions. If you let $$F:\mathbf {R\text - Alg}\to \mathbf {Ring}$$ be the functor that sends ...
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1answer
57 views

let $\dim(ker (A - \lambda I)) = 1$. why is $adj(A - \lambda I) \ne 0$

Let $\lambda$ is eigenvalue of $A$ and $\dim(\ker (A - \lambda I)) = 1$.($\lambda$ has geometric multiplcity one) why is $\text{adj}(A - \lambda I) \ne 0$?
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1answer
31 views

Quadratic Functional Differentiability

I would like to solve the following: Let $T$ be a self-adjoint bounded operator on a Hilbert space $H$. Consider the quadratic functional $\Phi$ defined by: \begin{equation} ...
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2answers
34 views

relation eigenvalue and adj(A-λI)

Let $A$ be a matrix in $\mathbb C^{n×n}$, let $λ$ be an eigenvalue of $A$ with eigenvector $x$. Why is there some $y \in \mathbb C^n$ such that $adj(A−λI)=x{y^*}$?
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1answer
48 views

Showing that if $A$ is closed, then $A^\ast A$ is self-adjoint

Let $A$ be a closed linear operator on a Hilbert space $H$. Then I want to show that $B = A^\ast A$ is self-adjoint. Now, $B$ is positive, i.e. $\langle f, B f \rangle \geq 0 \forall f \in D(B)$. ...
0
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1answer
29 views

Find a general 2x2 matrix where A = adj(A)

I know how to find the adjoint of 2x2 matrix but I'm at a loss for finding a general 2x2 matrix where A = adj(A). Thanks for your help!
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4answers
69 views

Calculate inverse of matrix

If $$A=\begin{bmatrix} -5 & 1 & 0 & 0\\ -19 & 4 & 0 & 0\\ 0 & 0 & 1 & 2\\ 0 & 0 & 3 & 5\\ \end{bmatrix}, $$ how do I calculate $A^{-1}$? Is there any ...
0
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1answer
25 views

Adjoint operators in Hilbert space

Consider the linear and bounded operators $X$ and $Y$on a Hilbert space $\mathcal{H}$ with inner product $\langle \cdot,\cdot \rangle$. How can I show that $$ \langle XY \boldsymbol{v}, ...
2
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1answer
112 views

Integration by parts to find the adjoint operator

On the interval $(0,1)$ consider the differential operator $Lu=u''''+u'$ with boundary conditions $u(0)+u'(1)=u(1)+u'(0)=0$ $2u(0)+u''(1)=2u(1)+u''(0)=0$ $(1)$ I want to find the adjoint ...
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0answers
17 views

prove operator is normal

Consider, $$A = \begin{bmatrix} 0 & I \\ c^2\frac{\partial^2}{\partial x^2} &-c_d I \end{bmatrix}$$ I want to construct an inner product space where AA* = A*A. That is A is a normal ...
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2answers
41 views

Bounded Linear Operator and the Adjoint

Let $S$ be a linear operator with dense domain $\mathcal{D}(S)$ in the Hilbert space $\mathcal{H}$. Assume that the domain $\mathcal{D}(S)$ belongs to a larger domain, namely $\mathcal{D}(S) \subset ...
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0answers
56 views

Functors adjoint from both sides

If F is left adjoint to G and G is left adjoint to F, does it imply anything nice about these functors?
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1answer
38 views

Question about the notation $S \subset T$ ,where $S$ and $T$ are operators

I want to prove that if $S\subset T$. Then $T^{*}\subset S^{*}$. But what does $S\subset T$ mean? $S$ and $T$ are operators and not sets.. :/
2
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3answers
88 views

Adjoint operator of $L^\infty$

Lets denote with $(\Omega,\Sigma,\mu)$ a $\sigma$-finite measurble space with a linear, continuous operator $$T : L^\infty \to L^\infty.$$ Does this always imply the existence of a linear, continuous ...
3
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1answer
35 views

Universality of tensor product from monoidal structure

As a follow-up to this previous question of mine, I'm trying to understand how to obtain tensor products from internal homs. I'm having a lot of difficulties and have found myself stuck already in ...
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1answer
30 views

A connection between a matrix norm and a related matrix's largest eigen-value

I have been asked to prove that for $A\in M_n(\mathbb{C})$, with $||A||:=\sup_{x\in\mathbb{C}^n,|x|=1}|Ax|$, $$||A||=\sqrt{\lambda}$$ where $\lambda$ is the eigen value of largest modulus of $A^*A$. ...
2
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1answer
31 views

adjoint representations

I am trying to work out the adjoint representations of $$H=\left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right), X = \left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} ...
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0answers
272 views

Adjoint of an adjoint linear map

My question is as it says in the title really. I've been reading Nakahara's book on geometry and topology in physics and I'm slightly stuck on a part concerning adjoint mappings between vector ...
2
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1answer
63 views

A functor that has both left and right adjoints

What can we say about a functor that has both left and right adjoints? I vaguely recall hearing that it is then an equivalence of category. Is it true? If not, then under what conditions it is true? ...
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0answers
24 views

Proof of inverse matrix element explicit formula [duplicate]

There is a matrix: A, and exists an inverse matrix: A^-1 which elements are b. (b)ij = adj(Aji) / det(A) What is the proof of this equation?