Intereting Posts

Rational + irrational = always irrational?
Proving a proposition which leads the irrationality of $\frac{\zeta(5)}{\zeta(2)\zeta(3)}$
Multivariate Taylor Series Derivation (2D)
trouble calculating sum of the series $ \sum\left(\frac{n^2}{2^n}\right) $
Probable prime test for specific class of $N=k \cdot b^n-1$
Showing that $\int_0^\infty \frac{|\cos x|}{1+x} \, dx$ diverges (Baby Rudin Exercise 6.9)
Prove: $\lim\limits_{n \to \infty} \int_{0}^{\sqrt n}(1-\frac{x^2}{n})^ndx=\int_{0}^{\infty} e^{-x^2}dx$
Prove or disprove statements about the greatest common divisor
Is it possible to generate a unique real number for each fixed length sequence of real numbers?
Conditional expectation of book Shiryaev page 233
Arrow's Impossibility Theorem and Ultrafilters. References
Derivation of the quadratic equation
Is this a Delta Function? (and Delta as limit of Gaussian?)
Self-teaching myself math from pre-calc and beyond.
crazy problem – does it have a solution? number theory perhaps?

Let $H$ be a $\mathbb R$-Hilbert space and $A$ be a bounded linear operator on $H$. If $A$ is nonnegative and self-adjoint, then there is a unique nonnegative and self-adjoint bounded linear operator $B$ on $H$ with $B^2=A$. This can be proved by an elementary version of the spectral theorem.

Now, suppose that $A$ is only nonnegative (and not necessarily self-adjoint). I’ve read that we can still find a unique nonnegative bounded linear operator $B$ on $H$ with $B^2=A$. How can we prove this statement?

My knowledge in operator theory is rather limited. So, I hope we can provide an elementary proof.

- Proving that a Hölder space is a Banach space
- Given a $C_c^∞(G)$-valued random variable, is $C_c^∞(G)∋φ↦\text E$ an element of the dual space of $C_c^∞(G)$?
- How to prove this inequality in Banach space?
- Analytic Vectors (Nelson's Theorem)
- The space of continuous functions $C()$ is not complete in the $L^2$ norm
- Continuous inclusions in locally convex spaces

- Weak topologies and weak convergence - Looking for feedbacks
- How to deduce open mapping theorem from closed graph theorem?
- Examples of statements that are true for real analytic functions but false for smooth functions
- Limit of sequence of growing matrices
- Fundamental Theorem of Calculus for distributions.
- Weak limit and strong limit
- Equicontinuity if the sequence of derivatives is uniformly bounded.
- a compact operator on $l^2$ defined by an infinite matrix
- The adjoint of finite rank operator is finite rank
- How common is it for a densely-defined linear functional to be closed?

- Is $\mathrm{ZFC}^E$ outright inconsistent?
- When a congruence system can be solved?
- Probability that the convex hull of random points contains sphere's center
- Eigenvalues of the principal submatrix of a Hermitian matrix
- Limit of $\lim_{x\rightarrow 1}\sqrt{x-\sqrt{x-\sqrt{x-\sqrt{x-…}}}}$
- Can you prove a random walk might never hit zero without the countable additivity axiom?
- Proving that an integer is even if and only if it is not odd
- Hyperbolic metric geodesically complete
- What kind of book would show where the inspiration for the Laplace transform came from?
- For what values of $\alpha,\beta$ is $x^{\alpha}\sin{x^\beta}\in L^1((0,1])$?
- Proof that the product of two differentiable functions is also differentiable
- How to find the integral $\int \tan (5x) \tan (3x) \tan(2x) \ dx $?
- Number of Solutions to a Diophantine Equation
- Why is $\det(A – \lambda I)$ zero?
- Show that $f(x+y)=f(x)+f(y)$ implies $f$ continuous $\Leftrightarrow$ $f$ measurable