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### Example 4.5-3 in Kryeszig's Functional Analysis: What is the relation between the matrix of a linear operator and that of its adjoint?

Let $X$ and $Y$ be finite-dimensional normed spaces, either both real or both complex, and let $T \colon X \longrightarrow Y$ be a linear operator. (Then by Theorem 2.7-8 in Kreyszig $T$ is bounded ...
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### If $D$ be the differentiation operator on $V$. Find $D^*$.

Let $V$ be the vector space of the polynomials over $R$ of degree less than or equal to $3$ with the inner product space $(f|g)=\int_{0} ^{1}f(t)g(t) dt$, and let $D$ be the differentiation operator ...
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### Find the operator norm: $T\in B(L^{2}(\mathbb{R},e^{-x^{2}}dx))$, $Tf(x) = f(x+1)$. $||T|| =$?

Let $L_{e}(\mathbb{R})$ denote the Hilbert space with inner product $\langle f, g \rangle = \int_{-\infty}^{\infty}f(x)\overline{g(x)}e^{-x^2}dx$. Let $T: L_{e}(\mathbb{R}) \to L_{e}(\mathbb{R})$ be ...
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### Given $T:E^*\to E^*,$ does there exist $S:E\to E$ such that $\langle Te^*,e \rangle = \langle e^*,Se\rangle?$

Let $E$ be a Banach space and $T:E\to E$ be a bounded linear operator. Denote $E^*$ a dual space of $E,$ that is, the space of all bounded linear functional $e^*:E\to\mathbb{R}.$ It is well known ...
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### Prove $||T^n||=||T||^n$ if $T\in B(H)$ is self-adjoint

Let $H$ be a complex Hilbert space and let $T\in B(H)$ be self-adjoint. I have already proven that $||T^{2^k}||=||T||^{2^k}$ for $k=0,1,2,\dots$. Now I have to prove the more general statement that ...
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### What is the intuition behind the kernel of $A$ being orthogonal to the range of $A^*$?

We know that, if $A$ is a linear bounded operator, then $\operatorname{Ker}(A) \perp \operatorname{Range}(A^*)$, where $A^*$ is the adjoint of $A$. I have no troubles understanding the proof of this (...
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### self-adjoint operator, such that $T^2=T^3,$ is orthogonal projection.

Let $T$ be a self-adjoint operator on a Hilbert space $H$, such that $T^2=T^3.$ Prove that $T$ is an orthogonal projection. Is that true if we had $T^4=T^3$? Thank you for the help.
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### 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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### 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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### Definition of the formal $L^2$-adjoint $T^*$ of a linear operator $T:C^\infty(T^*M\odot T^*M)\to C^\infty(M)$

Let $(M,g)$ be a Riemannian manifold, $C^\infty(T^*M\odot T^*M)$ the space of all smooth symmetric $2$-tensor fields on $M$, and $C^\infty(M)$ the space of all smooth functions on $M$. I'd like to ...
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### Counter example for infinite dimensional vector space

Consider a linear operator O acting on a Hilbert space H. If the dimension of H is finite, I have shown that: dim(ker O) = dim(ker O†). Where O† is the Hermitian conjugate of O. Does this hold when ...
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### Example for trivial intersection of domains

For a bounded operator $A$ in a Hilbert space $\mathcal H$ the real part of $A$ is defined by $\operatorname{Re}(A) = \frac 12(A+A^*)$. However, if $A$ is unbounded, this operator is defined on the ...
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### Unitary operators, inner products and unit operators with Dirac notation

From my notes, I have that for a unitary matrix: $$\underline{\underline{U}}^\dagger=\underline{\underline{U}}^{-1},\qquad \underline{\underline{U}}^\dagger=\big(\underline{\underline{U}}^{T}\big)^*$$ ...
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### Is the kernel of the adjoint operator equal to the kernel of the operator ($\ker (A)=\ker (A')$)?

I am in a middle of a proof where I asked myself about the following: Is the kernel of the adjoint operator equal to the kernel of the operator ($\ker (A)=\ker (A')$)? Theorem:Let $X,Y$ be Banach ...
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### Reference request for the following theorem in Von Neumann algebras.

I know the following result is true for a von Neumann algebra $A$: Theorem: Let $A$ be a von Neumann algebra. Then every self-adjoint element $h\in A$, i.e $h=h^*$, is the limit of a sequence of ...
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