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 $$nu(T):=sup_{operatorname{codim} M<infty}inf_{xin M, ,|x|=1}|Tx|$$ 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:

Quadratic formula, operators included here:
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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