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

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How to write a non-homogeneous equation in self-adjoint form

How can I write a non-homogeneous equation in self-dajoint form? such as, for equation with $-1\le x \le1$ $$(1-x^2)u''-xu'+2u=x^4+x$$ What is its self-dajoint form? Also, for a ...
2
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1answer
22 views

Role of metric in the matrix representation of Hermitian adjoint

I'm working through Jeevanjee's "An Introduction to Tensors and Group Theory for Physicists", and while trying to prove that the matrix representation $M(A^\dagger)$ of a Hermitian adjoint $A^\dagger$ ...
2
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2answers
65 views

If $\|Tv\|=\|T^*v\|$ for all $v\in V$, then $T$ is a normal operator

I have solved a question but I am not sure the last step of the question. If someone can verify it that would be great. Let $V$ be a finite dimensional vector space with complex inner product. Let ...
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1answer
44 views

Strong epimorphic counit iff conservative right adjoint?

On page 13 of Lack and Street's Combinatorial Categorical Equivalences, it is written (but not proven) that: A right adjoint is conservative if and only if the components of the counit are strong ...
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0answers
66 views

Can I assign a distinct homomorphism to every function?

I consider an algebraic structure and particular structure preserving morphisms (think groups and group homomorphism). I wonder if I can assign a distinct such morphisms to every function between any ...
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1answer
54 views

How are the cardinalities of the object images of adjoint functors related?

Here is a very silly question: Adjoint functors satisfy $$\mathrm{hom}_{\mathcal{C}}(FA,B) \cong \mathrm{hom}_{\mathcal{D}}(A,GB).$$ I consider numbers $a,b$ and read this as ...
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1answer
57 views

Proof that left adjoints preserve direct limits

I am reading Rotman's book on Homological algbra and have a slightly different proof of the statement in the title of this question. Am writing my attempt below. Could someone please advise me if I am ...
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1answer
109 views

Adjoint Functor Theorem

The Freyd's Adjoint Theorem states that given a complete locally small category $\mathcal{C}$, a continuous functor $G: \mathcal{C} \to \mathcal{D}$ has a left adjoint if and only if it satisfies a ...
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1answer
35 views

What is the dual of $A\cap B$

I encountered with some elliptic problem which admits a variational formulation in terms of space $X$ and I need to understand its dual. Suppose that $2<p<\infty$, $\Omega\subset {\mathbb R}^d$ ...
2
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1answer
21 views

Proving that the transformation obtained from an adjoint pair is natural

I am reading Homological Algebra by J.J. Rotman and am unable to do this problem. Given an adjoint pair $(F,G)$ where $F : \mathcal{C} \to \mathcal{D} $ and $G : \mathcal{D} \to \mathcal{C} $ are two ...
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2answers
44 views

Basic Criterion on Selfadjointness

How to prove the following in a neat subjective way: $$A\text{ symmetric}:\quad\mathcal{R}(A\pm\imath)=\mathcal{H}\implies A^*=A$$
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2answers
27 views

Proof: adjoint map of projection is a projection and …

Let $V$ be a pre hilbert space and $\pi \in \mathrm{End}(V)$. Show: the adjoint map $\pi^+$ of a projection (meaning: $\pi^2 = \pi$) is a projection itself. Show then: a projection $\pi$ is ...
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2answers
18 views

nilpotent linear transformation and its adjoint exercise

The problem statement. Let $(V,<,>)$ be a finite dimensional vector space equipped with an inner product, whose dimension is $n$, and let $f:V \to V$ be a nilpotent linear transformation such ...
4
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1answer
32 views

Using inner product property to determine if operator is an isomorphism.

Let $\varphi$ be an operator on a $k$-vector space $V$ with an inner product $\langle\cdot,\cdot\rangle$. Suppose that $\langle v,\varphi v\rangle = 0$ for every $v\in V$. If we take $k=\mathbb R$, is ...
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0answers
47 views

Closure operators and complete lattices.

A closure operator on a set $A$ is a function $C: \mathcal{P}(A) \to \mathcal{P}(A)$ satisfying following axioms: $X ⊆ Y \implies C(X) ⊆ C(Y)$ $X ⊆ C(X)$ It may also satisfy some additional ...
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2answers
26 views

help me please about adjoint of operators in L1

A : L₁→L₁ 1) A x=( x₁, x₂,.....xn , 0,0,....) 2) A x= (λ₁ x₁ ,λ₂ x₂,.....) |λ n|≤1 and λ n ∈ R I need to find adjoint of operators A in given space. ...
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1answer
29 views

help,example about disjoint operators

$T\colon L^2[0,1]→L^2[0,1]$ is given by $$ Tx(t)=∫_0^1 tx(s)\,ds $$ How can we find adjoint operator of $T$ in this space? $\langle Tx,y\rangle= \langle x,T^*y\rangle$ should be okay.But what ...
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2answers
97 views

Right adjoint to forgetful functor $\mathbf{Cat} \to \mathbf{Graph}$

There is a forgetful functor $U:\mathbf{Cat} \to \mathbf{Graph}$, which assigns a (small) category to its underlying (small) graph. Also, it has a left adjoint $F:\mathbf{Graph} \to \mathbf{Cat}$, ...
2
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1answer
65 views

Adjoint of an integral operator

I'm reading through a text about integral operators and I've come across the following theorem: Let $k:\mathbb{R}^2\rightarrow\mathbb{C}$ be a kernel, $T:L^2(\mathbb{R})\rightarrow ...
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3answers
104 views

adj (A*B) = adj B * adjA

How can I prove that adj (A*B) = adj B * adj A, if one of them (or both) singular matrix. I know how to prove it for non singular matrices, but I have no idea what to do in this case.
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1answer
83 views

Is it possible to define an inner product such that an arbitrary operator is self adjoint?

Given a vector space $V$ (possibly infinite dimensional) with inner product $(.,.)$. We say an operator $A$ is self adjoint if $(Af,g)=(f,Ag)$. The definition as stated require us to start with an ...
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1answer
28 views

Show $\mathrm{rank}\mathsf{T} = \mathrm{rank}\mathsf{T}^\ast$ for a linear operator of finite-dimensional inner product space

I need to show that $\mathrm{rank}\mathsf{T} = \mathrm{rank}\mathsf{T}^\ast$ for a linear operator $\mathsf{T}$ on a finite-dimensional inner product space $\mathsf{V}$. Let $\beta = ...
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2answers
65 views

Duality 2-functor on adjunctions

This question is about the definition of the duality 2-functor in Hovey's book on Model categories, Section 1.4. There he defines the 2-category of categories with adjunctions as follows: objects ...
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1answer
36 views

existence of an “inverse” adjoint in Banach Spaces

Let X,Y be Banach spaces and S a bounded operator $S: Y' \rightarrow X'$ where $'$ denotes the dual space or the adjoint operator depending on what it is on. Then $$ \exists \ \ T \in B(X,Y) : T'=S ...
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1answer
20 views

Finding the Hilbert Adjoint in this case

If we let $H$ be a Hilbert space with inner product $\langle.,.\rangle$. And we fix $y, z \in H$. Then let $T:H\rightarrow H$ be the bounded linear operator $Tx = \langle x,y\rangle z$. Then what is ...
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1answer
27 views

Singular matrices over a commutative ring $R$, with a given adjoint matrix

First, I apologize if this is a duplicate question. I also must apologize if this has a trivial solution. This question has two parts: Let $R$ be a commutative ring with $1$, and let $F = R^n$ be a ...
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0answers
28 views

How to evaluate this covariance that is given by a Gram matrix in Hilbert space

Define the linear operator $O_T: \mathbb{R}^n \to L_2[0,T] $ \begin{equation} O_Tx = Ce^{At} x, \end{equation} where $t \in [0, T]$, $ C$ and $A$ are matrices with compatible dimentions, and $x \in ...
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0answers
21 views

Find the covariance of estimation error

Define the linear operator $O_T: \mathbb{R}^n \to L_2[0,T] $ \begin{equation} O_Tx = Ce^{At} x, \end{equation} where $t \in [0, T]$, $ C$ and $A$ are matrices with compatible dimentions, and $x \in ...
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31 views

Relationship between eigenvalues of differential operator and eigenvalues of its adjoint operator.

I am considering $L\phi = -\triangle \phi + u \cdot \nabla \phi$ and its "adjoint" operator $L^* \phi = -\triangle \phi - \nabla \cdot (\phi u)$ on a bounded domain $\Omega \subseteq \mathbb{R}^n$. ...
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2answers
76 views

Isometry <=> Adjoint left inverse [duplicate]

Is it true that: $$T\text{ isometric}\iff T^*\text{ left inverse}$$ Obviously: $$\text{"}\Rightarrow\text{": }\langle x,\tilde{x}\rangle=\langle Tx,T\tilde{x}\rangle=\langle x,T^*T\tilde{x}\rangle$$ ...
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2answers
65 views

Isometric <=> Left Inverse Adjoint

Is it true that: $$T\text{ isometric}\iff T^*\text{ left inverse}$$ Obviously: $$\text{"}\Rightarrow\text{": }\langle x,\tilde{x}\rangle=\langle Tx,T\tilde{x}\rangle=\langle x,T^*T\tilde{x}\rangle$$ ...
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1answer
59 views

Finding Matrix adjoint

Need someone to check my reasoning as I don't feel confident in this topic: consider $P_2$(C) with inner product $$<p(x), q(x)> = \int {q(x)p(x) dx} $$ T is defied by T(p(x)) = p'(x) + p(x) ...
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1answer
55 views

Condition for a reflective subcategory of a cartesian closed category to be an exponential ideal

Here's the question I think I'm asking, with background below if necessary: Question: The reflector $L$ left adjoint to the inclusion of a reflective subcategory $\mathcal L\to\mathcal E$ is ...
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1answer
45 views

Proof that sheafification induces isomorphism on stalks using adjoints

Let $\mathcal{F}$ be a presheaf on some topological space $X$. It is not hard to prove directly that the map $\mathcal{F}\rightarrow \mathcal{F}^{sh}$ induces an isomorphism of stalks (Here ...
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25 views

Self-Adjointness of beam stiffness operator

I think it is well known that the operator $\frac{EI}{\rho} \frac{\partial^4}{\partial x^4}$ which arises from a standard Euler-Bernoulli beam is self-adjoint in $H$, where $H = L^{2}$, given ...
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2answers
94 views

Are these adjoint functors to/from the category of monoids with semigroup homomorphisms?

Do the forgetful functors $G_H:\bf Monoid \to \bf Semigroup^1$ and $G_O:\bf Semigroup^1 \to \bf Semigroup$ have left and/or right adjoints? Here $\bf Semigroup$ is the category of semigroups with ...
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2answers
72 views

Checking whether an operator is self-adjoint. Problem with domain of an operator.

I want to check whether the position operator $A$, where $Af(x)=xf(x)$ , is self-adjoint. For this to be true it has to be Hermitian and also the domains of it and its adjoint must be equal. The ...
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1answer
25 views

Composition of Functors which have adjoints has also an adjoint [duplicate]

The exercise is the following: Suppose $F$ has right adjoint $G$ and $H$ has right adjoint $J$ ($F:\mathbb{A}\rightarrow\mathbb{B}$ and $H:\mathbb{B}\rightarrow\mathbb{C}$ with $\mathbb{A,B,C}$ ...
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2answers
53 views

Adjoint to a functor $\textbf{PoSets}\rightarrow\textbf{PreOrd}$

i have a question about adjoints in category Theory. Let $\textbf{Posets}$ the category of Posets (thus Sets with binary relation $\leq$ which is reflexiv, transitiv and antisymmetric) and let ...
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2answers
80 views

Normal Self-Invertible Operator is Self-Adjoint

If $T\in B(H)$ for some Hilbert space $H$, is a normal operator and $T^2=I$, then $T=T^*$. It seemed simple when I first saw the claim, but I'm having trouble showing it. I know that it implies ...
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2answers
83 views

Equivalence of the definition of Adjoint Functors via Universal Morphisms and Unit-Counit

I was reading here about adjoint functors, and I was following the construction of the right adjoint to a left adjoint functor, and I kept getting tripped up over showing that the resulting functor ...
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1answer
61 views

Adjoint to identity functor must be trivial

If an endofunctor F on some category $\mathfrak{C}$ is (left) adjoint to the identity functor.... then does it necesarilly have to also be the identity functor... Since $\mathfrak{C}(F(X),Y)\cong ...
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Adjoint boundary problem

Suppose $A\subset\mathbb{R}^2$ is open and bounded and has a boundary $\partial A$ which is $C^1$-smooth. Let $N$ be the outward normal unit vector field of $A$ and $T$ the counterclockwise pointing ...
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1answer
29 views

Adjoints to cofree modules tensor?

If $M$ is a cofree $R$-module and $A,B$ are arbitrary $R$-modules then, is there a left adjoint to the functor $M\otimes_R -$, i.e. is there an endofunctor $F$ on $_R \mathrm{Mod}$ such that ...
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1answer
71 views

Composition of adjoint functors

Does the composition of adjoint functors again form an adjunction? Say $\langle F_1,G^1\rangle$ is an adjunct pair between two categories A and B and $\langle F_2,G^2\rangle$ is also an adjoint pair ...
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2answers
55 views

$\mathbb{Q}$ adjoining primes and the sum of root of those primes

I have $p$, $q$ as primes, and I want to show that $\mathbb{Q}(\sqrt{p},\sqrt{q})=\mathbb{Q}(\sqrt{p}+\sqrt{q})$. I was thinking about using inclusion both ways, so what does an element in ...
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1answer
56 views

Show that $T\neq{T^*}$

Let $V=P_2(\mathbb{R}), T\in \mathcal{L}(P_2(\mathbb{R})),$ where $T(p)=(a_1x)$. Make $V$ an inner product space by defining $$\langle p,q\rangle=\int_0^1{p(x)q(x)\,dx}$$ So I calculate $$\langle ...
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1answer
60 views

Show a linear transform is self adjoint - check my answer

We are given $T:V \to V$ a normal linear transform (meaning $TT^*=T^*T$) We are also given $T^2=T$. Show that $T$ is self adjoint (meaning $T^*=T$). What I did I think I may have done something ...
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2answers
88 views

Why are left/right adjoint functors not called up/down?

I am studying category theory and I recently learned about adjoint pairs of functors. It seems to me that they are called left and right adjoints because if we have categories $\mathcal{C}$ and ...
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Adjoint of unbounded Operators: Product and Sum

When precisely does equality hold for sum and product: $$S^*+T^*\subseteq (S+T)^*$$ $$S^*T^*\subseteq (TS)^*$$ So far I checked that for the sum the seemingly weaker condition implies the stronger ...