Intereting Posts

What is the metric tensor on the n-sphere (hypersphere)?
Is it a new type of induction? (Infinitesimal induction) Is this even true?
Truncated alternating binomial sum
Show that if $G$ is simple a graph with $n$ vertices and the number of edges $m>\binom{n-1}{2}$, then $G$ is connected.
Can the semidirect product of two groups be abelian group?
A set that it is uncountable, has measure zero, and is not compact
Semi-Norms and the Definition of the Weak Topology
Can a number have infinitely many digits before the decimal point?
Expected number of runs
A ‘strong’ form of the Fundamental Theorem of Algebra
Trig Fresnel Integral
Absolute and uniform convergence of $\sum _{n=1}^{n=\infty }2^{n}\sin \frac {1} {3^{n}z}$
How can I show that $f$ must be zero if $\int fg$ is always zero?
How to solve the equation $n^2 \equiv 0 \pmod{584}$?
How can I pack $45-45-90$ triangles inside an arbitrary shape ?

Let say we are in a complex vector space, is there an example of a normal operator with only real eigenvalues(or without eigenvalues) that is not a self-adjoint operator? Cause of the spectral theorem it is impossible for the finite dimensional case. I have no idea in the infinite case. I would appreciate any help! Thanks!

- Proof/Intuition for Eigenvalues to Solve Linear Differential Equations
- Show $\alpha$ is selfadjoint.
- Show that when $BA = I$, the solution of $Ax=b$ is unique
- Are vector spaces and their double duals in fact equal?
- Proof of Uncountable Basis for $\mathbb{N} \to \mathbb{R}$ over $\mathbb{R}$
- Given a $4\times 4$ symmetric matrix, is there an efficient way to find its eigenvalues and diagonalize it?
- Analogy between linear basis and prime factoring
- Homogeneous polynomials on a vector space $V$, $\operatorname{Sym}^d(V^*)$ and naturality
- Does the inverse of a polynomial matrix have polynomial growth?
- Under what conditions is a linear automorphism an isometry of some inner product?

To elaborate on fedja’s comment: Let $(X,\mu)$ be a measure space, let $h$ be a bounded measurable complex-valued function on $X$, and let $T$ be the multiplication operator on $L^2(X,\mu)$ defined by $Tf = hf$. Show that $T$ is normal, and is self-adjoint iff $h$ is real-valued almost everywhere. Now show that $\lambda$ is an eigenvalue of $T$ iff $\mu(\{h= \lambda\}) > 0$. Taking as an example $X = [0,1]$ with Lebesgue measure, you should be able to use this to construct a normal, non-self-adjoint operator with only real eigenvalues, or with no eigenvalues at all.

- An integrable Functions is almost everywhere finite
- Show that $\mathfrak{S}=\bigcup_{N=1}^{\infty}\mathfrak{Z}_N\cup\left\{\emptyset\right\}$ is a semi-ring
- A category of relations – or two different?
- What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring?
- complex polynomial satisfying inequality
- Microeconomics (quasi-linear utility function)
- Uspenskij-Tkachenko Theorem
- Solution to congruence $z^2=c$ mod n
- Is there more than one way to express a derivative as a limit of a quotient?
- Compute $\int_{0}^{\infty}\frac{x\sin 2x}{9+x^{2}} \, dx$
- Does the sum of digits squared of a number when we keep on doing it lead to a single digit number?
- Are there any bases which represent all rationals in a finite number of digits?
- Long time behavior heat equation on infinite line
- Is a compact Hausdorff space metrizable? Maybe even complete?
- Unique Decomposition of Primes in Sums Of Higher Powers than $2$