Operator theory is the branch of functional analysis that focuses on bounded linear operators, but it includes closed operators and nonlinear operators. Operator theory is also concerned with the study of algebras of operators.

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Given a vector x, can we say something about an A such that A x = x?

Let us assume that a vector $x \in \mathbb{R}^n$ is given and we are looking for a matrix $A \in \mathbb{R}^{n\times n}$ which yields $A x = x$. That is, we perform a sort of reverse questioning: ...
1
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1answer
329 views

Self adjointness of square root operator

Theorem: If $A$ is self adjoint and nonnegative, then $A$ has a unique nonnegative square root $A^{\frac{1}{2}}$. As I understand, thesis of this theorem say only about the existence of ...
3
votes
1answer
87 views

Is $B - B'$ self-adjoint provided $B,B'$ are positive operators?

If I have two positive operators $B,B'$ on an arbitrary Hilbert space $\mathcal{H}$ not necessarily over $\mathbb{C}$, how do I know that $B - B'$ is self adjoint? EDIT: Reed and Simon define ...
3
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1answer
49 views

When does an operator commute with another operator given by a series?

Suppose $B$ is a bounded operator on some Hilbert space $\mathcal{H}$, given by a series of the form $$ B = I + \sum^\infty_{k = 1} c_k(I - A)^k $$ where $A$ is a given bounded operator on ...
6
votes
3answers
829 views

Books for studying Dirac Operators, Atiyah-Singer Index Theorem, Heat Kernels

I am interested in learning about Dirac operators, Heat Kernels and their role in Atiyah-Singer Index Theorem. From various sources (including this very helpful question), I have come to know of ...
3
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3answers
3k views

Commutator of $x$ and $p^2$

I have a question: If I have to find the commutator $[x, p^2]$ (with $p= {h\over i}{d \over dx} $) the right answer is: $[x,p^2]=x p^2 - p^2x = x p^2 -pxp + pxp - p^2x = [x,p]p + p[x,p] = 2hip$ But ...
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votes
1answer
20 views

Difference operator endomorphism

Let $\delta : R_{p}[x] \to R_{p}[X] $ the endomorphism of $R_{p}[X]$ such that : $\delta(P(X)) = P(X + 1) - P(X)$ , what is the kernel of $\delta$ ? (i tried to compute it explicitly but that was a ...
3
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6answers
568 views

Nilpotent linear operators

Suppose that $T : V \to V$ is a linear operator on an $n$-dimensional vector space $V$. (a) Show that for all $i$, $\ker T^i \subset \ker T^{i+1}$. (b) Show that if $\ker T^k = \ker ...
0
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2answers
91 views

$T:\ell_2\to \ell_2, T(x_1,x_2,\dotsc)=(x_1,x_2/2,x_3/3,\dotsc)$

$T:\ell_2\to \ell_2, T(x_1,x_2,\dotsc)=(x_1,x_2/2,x_3/3,\dotsc)$ I need to know whether it is self adjoint and unitary operator given that $x_i\in\mathbb C$ I am not able to do it please tell me how ...
3
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2answers
214 views

Is it a unitary, self adjoint and normal operator?

Let $A\colon H\to H$ be a bounded linear operator on a complex Hilbert space such that $\|Ax\|=\|A^*x\|\forall x$, given that there is a nonzero $x$ for which $A^*x=(2+3i)x$. Then I need to know ...
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vote
0answers
86 views

Show that $(D_t t)$ is an isomorphism.

Let $B$ be a Banach space, $\epsilon > 0$, and $$C_1^0 ([-\epsilon,\epsilon],B) = \{u \in C^0([-\epsilon,\epsilon],B) : tu \in C^1([-\epsilon,\epsilon],B)\}.$$ Denote $\partial/\partial t$ by ...
2
votes
1answer
64 views

A question about quotient space of $R(T^{n})$

I am reading a paper about spectral theory. The author says it is easy to see the following proposition: For $T\in L(X)$, if dim$(R(T^{d})/R(T^{d+1}))<\infty$, then $R(T^{d})$ is closed if and ...
1
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1answer
66 views

Are $T,T^2$ compact operators?

$T:l_2\to l_2$ is defined by $T(x_1,x_2,\dots)=(0,x_1,0,x_3,0,x_5,\dots)$ we need to find whether $T, T^2$ are compact or not. I see here the definition of compact operator but I'm not able to apply. ...
4
votes
1answer
124 views

Can 0 be an eigenvalue?

Let $-\Delta $ be the positive Laplacian and consider the operator $$ -\Delta + V $$ on $L^2(\mathbb{R}^3)$ with domain the Sobolev space $W^{2,2}(\mathbb{R}^3)$. Here $V:\mathbb{R}^3\to \mathbb{R}$ ...
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1answer
54 views

Question about operators on Hilbert space

Let $\cal{H}$ be a Hilbert space, $P_1,P_2,\cdots,P_m$ a sequence of orthonormal projections such that $P_iP_j=0$ for $i\neq j$ and $P_1+P_2+\cdots+P_m=I$. Then $\|\sum^m_{k=1}P_kTP_k\|\leq\|T\|$ for ...
4
votes
1answer
127 views

Decomposing operators into the sum of a quasinilpotent and something else

I seem to remember some result of the following sort: Alleged Theorem. Every bounded operator on a separable complex Hilbert space can be decomposed as the sum of a normal operator and a ...
2
votes
1answer
99 views

An quasi-nilpotent operator restricted to a subspace is a nilpotent?

I am reading a paper about operator theory, there is a proposition I could not understand. Let $T\in L(X)$ be a quasi-nilpotent operator and $X_{1}$ be a non-zero finite-dimensional subspace of X, ...
2
votes
2answers
83 views

Maximum of two positive operators

Let $A,B$ be two positive operators in $B(H)$. Does there exist, in general, an operator $C$ such that for each $T$, if $A \leq T$ and $B \leq T$, then $$A\leq C \leq T\quad \text{and}\quad B\leq ...
5
votes
2answers
111 views

Linear operators on $C^\infty[a,b]$

I do not know too much about linear operators so forgive me if this doesn't make much sense, but what would the space of linear operators on $C^\infty[a,b]$ be defined as? If we denote this space as ...
4
votes
1answer
268 views

How to determine a operator norm?

How to solve following: In Hilbert space $W_2^1=\{f:[0,1]\rightarrow \mathbb{C}|f\in AC[0,1], f'\in L^2[0,1]\}$ with scalar product $(f,g)=\int_0^1 f\overline{g}dx+\int_0^1 f'\overline{g'}dx$ is ...
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2answers
232 views

Operator Theory References and Topics

I wish to do a reading course in Operator Theory. Thus, I am looking for some references in the area. Right now, I have the following two sources available: Unbounded Self-Adjoint Operators ...
2
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1answer
72 views

When does Gelfand Naimark Theorem Hold?

I was going through the proof of the Gelfand Naimark Theorem for the Unital Commutative Banach Algebras. In proving that each character has norm equal to $1$, we used the fact that $\|e\| = 1$ where ...
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1answer
73 views

Intuition concerning Schwartz kernels of Operators

Consider a (for example differential) operator $A$ acting on an appropriate function space over a smooth compact manifold without boundary. Using the Schwartz kernel $K(x,y)dy$ of the operator, its ...
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0answers
103 views

Transpose of the Hilbert-Schmidt operator

Let $X = L^2(\Omega)$, $\Omega \subset \mathbb{R}^N$ be an open set (or a $\sigma$-finite measure space), $B \in L^2( \Omega \times \Omega)$. Then the Hilbert-Schmidt operator $T \in \mathcal L(X)$ ...
3
votes
1answer
65 views

$B$ be an Infinite dimensional Banach Space and $T:B\to B$ be an continuos operator

Let $B$ be an Infinite dimensional Banach Space and $T:B\to B$ be an continuous operator such that $T(B)=B$ and $T(x)=0\Rightarrow x=0$ which of the following is correct? $T$ maps bounded sets into ...
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0answers
75 views

Kernel of an infinite dimensional operator

Can the Kernel of an infinite dimensional operator have $\dim=0$? I am thinking to the annihilation ($\hat E$) and creation ($\hat E^\dagger$) operators. Suppose, in fact, we have an infinite but ...
4
votes
1answer
122 views

Counterexample using counting measure

While proving that the norm of the mulplicative operator from $L^2(X) \to L^2(X)$ is the essential supremum of $|g|$ where $g \in L^\infty(X)$, I found that I need the $\sigma$-finiteness of the ...
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0answers
43 views

self adjoint linear operator and integration

is this formula correct ?? $$ \int_{-\infty}^{\infty} Lf(x)\delta (x-1)= \int_{-\infty}^{\infty} f(x)L^{\dagger}\delta(x-1) $$ here $ L $ is a linear operator and $ L^{\dagger}$ is its formal ...
12
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2answers
337 views

Shift Operator has no “square root”?

Consider the left shift operator $T : \ell^1(\mathbb N) \to \ell^1(\mathbb N) $ by $$T(x_1,x_2..... )=(x_2, x_3 ........),$$ and also the right shift operator $S : \ell^1(\mathbb N) \to \ell^1(\mathbb ...
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1answer
98 views

Left support of an operator on a Hilbert space

The left support $l(x)$ of an operator $x$ between Hilbert spaces $\mathbb{H}$ and $\mathbb{K}$ is defined as the smallest projection $e \in \mathfrak{B}(\mathbb{H})$ such that $ex=x$. The question ...
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1answer
42 views

a question on decreasing sequence of subspaces II

This is related to this question see here Let $V= \mathbb{Q}^{\mathbb{Z}}$, considered as an infinite dimensional linear space over $\mathbb{Q}$. And assume $W=\mathbb{Q}^F$ is a finite dimensional ...
4
votes
1answer
78 views

Reflexivity of $X \times Y$

I want to prove the following Theorem. Let $X,Y$ be reflexive. Then $X \times Y$ is reflexive. Here my try. Proof. Let $J_X, J_Y$ be the canonical injections of $X$ onto $X''$ and of $Y$ onto ...
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1answer
50 views

Help showing $\phi _k$ is a bounded linear functional

Let $V$ be the space of continuous functions on the interval $[-\pi , \pi]$ with the $L^2$ norm $$\lVert f\rVert_2=\left(\int_{-\pi}^\pi |f(t)|^2\mathrm dt)\right)^\frac{1}{2}$$ For $f$ in $V$, define ...
2
votes
1answer
77 views

Does projection onto a finite dimensional subspace commute with intersection of a decreasing sequence of subspaces: $\cap_i P_W(V_i)=P_W(\cap_i V_i)$?

Let $V$ to be an infinite dimensional linear space over some field $k$. (you can take $k=\mathbb{C}$, or further assume $V$ is a complex Hilbert space). And assume $W$ is a finite dimensional ...
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0answers
64 views

Almost everywhere analytic function

Suppose we have a measure space $\Omega$ and a function $m\in L^\infty(\Omega,\mathcal{B}(E))$, that is invertible for almost all $\theta\in\Omega$ Further assume, that we have an other function $G$ ...
2
votes
1answer
264 views

Adjoint of a multiplication operator

Let $B$ be the Banach space of continuous functions vanishing at infinity and defined on a locally compact Hausdorff space $X$. Given a continuous and bounded function $g$ on $X$, let $T$ be the ...
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2answers
80 views

$a^*a$ has a non-negative spectrum

I am learning $C^*$-algebra, especially I work on the proof of the Gelfand-Naimark theorem. In many books such as the one of Arveson, it looks that the following lemma is the key stone of the proof: ...
3
votes
1answer
174 views

Show that an operator is bounded (from Reed and Simon)

I am currently reading Reed and Simon's IV: Analysis of Operators, Volume 4 (Methods of Modern Mathematical Physics). I don't understand something they do in Theorem XIII.64. The problem is: Let $A$ ...
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1answer
124 views

A problem with linear operator in a Hilbert space

Let $(H,(\cdot,\cdot)_H)$ and $(Q,(\cdot,\cdot)_Q)$ two Hilbert separable spaces s.t $H\subset Q$ and let $B:H\to Q$ a bounded and linear operator. Let $\sigma,\tau\in H$ two fixed elements. My ...
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1answer
185 views

Is this proof correct? (left inverse and topologically complementary subsets)

I want to prove the following theorem: Theorem. Assume $T \in \mathcal L ( X, Y )$ is injective. The following statements are equivalent: $T$ admits a left inverse; Im($T$) is closed and ...
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1answer
432 views

When solving a linear differential equation by factoring the operator, how does one guarantee no solutions are lost?

I think the best way to make this question clear is with an example. Lets say we want to solve the differential equation $(\Delta^2 - \lambda^4)\phi=0,$ calculations are greatly simplified if we ...
5
votes
2answers
159 views

Bounded operators with prescribed range - part II

This is a continuation of the question bellow, in a more particular case. Bounded operators with prescribed range If $X$ is a separable Banach space and $Y$ is a closed, infinite dimensional ...
4
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1answer
169 views

Non strictly singular operators

Let $X$ be a separable Banach space and let $T:X\to X$ be a bounded operator that is not strictly singular. Can we always find an infinite dimensional, closed, and complemented subspace $Y$ of $X$ ...
10
votes
3answers
281 views

Why is such an operator continuous?

These two questions were in one question of a list of exercises. Let $E$ be a Banach space and $T : E \longrightarrow E^*$ be linear. If $\langle T(x),x \rangle \geq 0$ holds for all $x \in E$, ...
4
votes
1answer
181 views

Bounded operators with prescribed range

If $X$ is a seaparable Banach space and $Y$ is a subspace of $X$, not necessarily closed, can one always find an bounded operator with range $Y$? It is easy when $Y$ is closed and complemented, what ...
5
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3answers
376 views

Continuity of the adjoint map in various operator topologies

I am currently reading about operator topologies in the book "Methods of Modern Mathematical Physics: Functional Analysis" by Reed and Simon. In their treatment of the Hilbert space adjoint, a ...
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1answer
90 views

Is the inclusion map in the Sobolev embedding theorem a surjective map?

Let $W^{k,p}(\mathbb{R}^n)$ be the Sobolev space of all real valued functions on $\mathbb{R}^n$ whose first $k$ weak derivatives are in $L^p(\mathbb{R}^n)$. Assume that $$ \frac{1}{q} = \frac{1}{p} ...
4
votes
2answers
208 views

Question about finding the norm of a bounded linear operator

Let H be a Hilbert space. Suppose $(i_k)_1^\infty$ is a complete orthonormal sequence in H. Let $a_k \in \mathbb{C}$ for $k \in \mathbb{N}$. Assume there is a bounded linear operator $T:H \rightarrow ...
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0answers
93 views

exponential of an operator, all to a power

I saw this almost answered here: Exponential of the differential operator (it is the unaccepted answer) What I am looking to "solve" is $$ \sum_{j=0}^d\; \left( e^{\epsilon\,\partial_x} \right)^j ...
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1answer
61 views

operator over inner product

Let $V$ be a finite-dimensional inner product space over $\mathbb{R}$ and let $u,v \in V$ be given. Define a linear operator $u\otimes v: V \rightarrow V$ by $(u\otimes v)x=<v,x>u$, where ...