riemannian geometry – Principal eigenvalue of non self-adjoint elliptic operators on closed manifolds

Consider the elliptic operator $$Lu = – Delta u + langle nabla u , X rangle + c , u$$ acting on functions on a closed Riemannian manifold $$M$$. Here $$Delta$$ denotes the Laplace-Beltrami operator, $$X$$ is an arbitrary smooth vector field, and $$c geq 0$$ is a smooth function on $$M$$ which does not vanish identically. Does $$L$$ have a so-called principal eigenvalue’ $$lambda_1 > 0$$, whose corresponding (unique up to scaling) eigenfunction does not change sign?

A similar statement holds for smooth domains in $$mathbb{R}^n$$, as shown for instance in Evans’ PDE book, chapter 6. Moreover, in this paper it is sated that this fact is equivalent to the operator satisfying a maximum principle (which is indeed the case for the above $$L$$).

linear algebra – The Operator Equation $AB = lambda BA$ for Self-Adjoint Operators

Suppose that $$A$$ and $$B$$ are self-adjoint bounded linear operators on a Hilbert space and $$lambda in mathbb{C}$$. It turns out that if $$lambda notin {-1, 1}$$ then $$AB=lambda BA implies AB = BA = 0$$.

Does anyone know of any applications of this result?

oa.operator algebras – Lower bounds in the space of compact operators

Let $$H$$ be a separable Hilbert space, and $$K(H)$$ the corresponding space of compact operators. Consider the “unit sphere” $$S:={Tin K(H)|Tgeq 0text{ and }||T||=1}$$. Is it true that, given any pair of operators $$T_1,T_2in S$$, there exists another operator $$Tin S$$ such that $$Tleq T_1,T_2$$?.

fa.functional analysis – Operational quantities characterizing upper semi-Fredholm operators

An operator $$T:Xrightarrow Y$$ is said to be upper semi-Fredholm if its range is closed and its kernel is finite-dimensional. M. Schechter (1972) introduced a quantity $$nu(T):=sup_{operatorname{codim} M and proved that $$T$$ is upper semi-Fredholm if and only if $$nu(T)>0$$.

For a bounded subset $$A$$ of a Banach space $$X$$, let $$chi(A):=inf{epsilon>0: A$$ has a finite $$epsilon$$-net in $$X}$$. Then $$A$$ is relatively norm-compact if and only if $$chi(A)=0$$. For an operator $$T:Xrightarrow Y$$, let $$h_{mathrm{cb}}(T):=inf{chi(TD):chi(D)=1},$$ where the infimum is taken over all countable bounded subsets $$D$$ of $$X$$ with $$chi(D)=1$$.

M. González and A. Martinón (1995) noted that these two quantities are equivalent:

$$frac{1}{2}h_{mathrm{cb}}leq nuleq 2h_{mathrm{cb}}.$$

But they did not provide the proof. I need a detailed proof of this result.

functional analysis – Generalization of a result related to the theorem of Eckhardt-Young-Mirsky to operators of Hilbert-Schmidt

Yes $$d in mathbb N$$, $$A in mathbb R ^ {d times d}$$ is symmetrical and $$lambda_1 ge cdots lambda_r$$ denote the eigenvalues ​​of $$A$$, then it is easy to show that $$sum_ {i = 1} ^ r lambda_i = sup _ { substack {V le mathbb R ^ d \ dim V = r}} operatorname {tr} U ^ ast AU tag1 ,$$ where the columns of $$U in mathbb R ^ {n times r}$$ are an orthonormal basis of $$V$$.

Is there a generalization of this result to compact self-supporting operators $$A$$ on a $$mathbb R$$-Hilbert space $$H$$?

The trace on the right side of $$(1)$$ is the inner product $$langle AU, U rangle _ { operatorname {HS} (H)}$$ of $$AU$$ and $$U$$ in space $$operatorname {HS} (H)$$ operators Hilbert-Schmidt. So, I guess we could generalize this result by assuming $$A in operatorname {HS} (H)$$.

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quadratic forms – Do we include operators before the coefficient when applying algebraic formulas to equations?

I do some simple retraining on college algebra after a long time, but I have come to find some inconsistencies in my understanding of how to apply the formulas, do we include operators before the coefficient when applying algebraic formulas to equations?
I fight in particular with these two formulas:

2x²+14x-196 = 0, when applied to the formula (-b±√(b²- 4ac))/2a, gives (-14±√((14)²-4(2)(-196)))/2(2). The answer here is x = -14, x = 7 checked.

Note that the coefficient "c" must take the negative operator to give the answer.

However:
If we are consistent and say yes, operators must be included in formulas:
When using the formula (what is this algebraic formula called btw?) (a-b)² = a²-2ab+b², when applied to (2-3)², gives: 2²-2(2)(-3)+(-3)² = 4+12+9=25`. Which is clearly the wrong answer.

I'm struggling with this inconsistency, can anyone help me?

fa.functional analysis – Definition of Lyapunov exponents for compact operators

There is the following well-known result from Goldsheid and Margulis (see Proposition 1.3) on the existence of Lyapunov exhibitors:

Let $$H$$ be a $$mathbb R$$-Hilbert space, $$A_n in mathfrak L (H)$$ be compact and $$B_n: = A_n cdots A_1$$ for $$n in mathbb N$$. Let $$| B_n |: = sqrt {B_n ^ ast B_n}$$ and $$sigma_k (B_n)$$ denote the $$k$$e greatest singular value of $$B_n$$ for $$k, n in mathbb N$$. Yes $$limsup_ {n to infty} frac { ln left | A_n right | _ { mathfrak L (H)}} n le0 tag1$$ and $$frac1n sum_ {i = 1} ^ k ln sigma_i (B_n) xrightarrow {n to infty} gamma ^ {(k)} ; ; ; text {for all} k in mathbb N tag2,$$ then

1. $$| B_n | ^ { frac1n} xrightarrow {n to infty} B$$ for some non-negative and self-adhesive compacts $$B in mathfrak L (H)$$.
2. $$frac { ln sigma_k (B_n)} n xrightarrow {n to infty} Lambda_k: = left. begin {cases} gamma ^ {(k)} – gamma ^ {(k -1)} & text {, if} gamma ^ {(i)}> – infty \ – infty & text {, otherwise} end {cases} right } tag2$$ for everyone $$k in mathbb N$$.

question 1: I have seen this result in many textbooks, but I have wondered why it is stated this way. First of all, isn't it $$(2)$$ clearly equivalent to $$frac { sigma_k (B_n)} n xrightarrow {n to infty} lambda_i in (- infty, infty) tag3$$ for some people $$lambda_i$$ for everyone $$k in mathbb N$$ which in turn is equivalent to $$sigma_k (B_n) ^ { frac1n} xrightarrow {n to infty} lambda_i ge0 tag4$$ for some people $$mu_i ge0$$ for everyone $$k in mathbb N$$? $$(4)$$ seems to be much more intuitive than $$(3)$$, since no $$lambda_i$$, But $$mu_i = e ^ { lambda_i}$$ are precisely Lyapunov's exponents of the limit operator $$B$$. Am I missing something? The definition of $$Lambda_i$$ (which is equal to $$lambda_i$$) seems odd to me.

question 2: What is the interpretation of $$B$$? Usually I look at a discrete dynamic system $$x_n = B_nx_0$$. What is $$B$$ (or $$Bx$$) tells us about the asymptotic behavior / evolution of orbits?

fa.functional analysis – A question about the comparison of positive self-attendant operators

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