Let $E$ be a Banach lattice and $T_n\in\mathcal{L}^r(E)$ a sequence of regular operators such that $T_n$ converges strongly to $T\in\mathcal{L}^r(E)$. How to prove that $\left|T_n\right|$ converges strongly to $\left|T\right|$, $\left|\cdot\right|$ is the modulus of the operator. The result holds true for the operator norm convergence.
2025-01-13 07:53:42.1736754822
Strong convergence of regular operators and convergence of the modulus
83 Views Asked by metic https://math.techqa.club/user/metic/detail AtRelated Questions in FUNCTIONAL-ANALYSIS
- equivalent definitions of weak topology on a topological vector space
- Interpreting function notation?
- Dimension of $\ell^{1}$.
- Existence of an element in the infinite dimensional normed linear space?
- Confusing on lower semi continuous and its application in minimize problem
- Uniform and Compact Open Topology on spaces of maps from $\mathbb{R} \rightarrow \mathbb{R}$
- Trace Class: Relativeness
- Extension theorem for Sobolev spaces $W^{1,\infty}(\Omega)$: is there an elementary proof?
- Counterexample to $L^1$-boundedness of the maximal operator $f \mapsto f^\#$ with $f^{\sharp}(x):=\sup_{Q\ni x}\frac{1}{|Q|}\int_{Q}|f-(f)_{Q}|dy$
- Video lectures on Functional Analysis
Related Questions in CONVERGENCE-DIVERGENCE
- Proving whether the limit of a sequence will always converge to 0?
- If I take pre-images of an increasing subset of the image, do their measures converge to that of the range?
- Derivative of power series
- Derivative of power series with nonnegative coefficients
- Convergence in probability of random probability measures
- Show that $\sum_{j=1}^{\infty}\frac{\sqrt{j+1}-\sqrt{j}}{j+1}$ is convergent or divergent.
- Proof of Simple Limit Theorem
- If $\sum_{n=1}^\infty a_n$ converges, prove that $\lim_{n\to \infty} (1/n) \sum_{k=1}^n ka_k = 0$.
- Radius Of Convergence of the series
- $\sum_{j=3}^\infty \frac{1}{j(\log(j))^3}$ converges or diverges?
Related Questions in OPERATOR-THEORY
- Trace Class: Relativeness
- Given an operator $ * $ and it's inverse $ \setminus $ when does $ x \setminus y = x * \left( 0 \setminus y \right) $?
- Existence of operator with certain properties on a Banach space
- Why is $\sqrt{T^*T}$ self-adjoint?
- A restriction of a symmetric operator such that the range of (operator)+i is the same
- Definition of "Extension" of Bounded Linear Transformation
- Operator norm of positive operator.
- Resolvent set/operator
- Finite measure operator norm
- For what operators $A$ on a Hilbert space is the identity operator in the closure of the similarity orbit of $A$?
Related Questions in BANACH-LATTICES
- Separation Hahn Banach theorem in vector Banach lattice
- Equivalences for a Banach lattice
- Positiveness of Inverse Of Positive Operator Implies Lattice Isomorphism?
- Examples of Normalized Semigroup of Operators
- Banach lattice from a positive linear form on the dual
- General form of elements in the vector lattice (Riesz Space) generated by a vector space
- Division in Banach Lattice Algebra
- Positive cone of Banach lattice algebra
- "Representation Capacity" of Finite Lattice Ordered Modules
- Strong convergence of regular operators and convergence of the modulus
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Refuting the Anti-Cantor Cranks
- Find $E[XY|Y+Z=1 ]$
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- What are the Implications of having VΩ as a model for a theory?
- How do we know that the number $1$ is not equal to the number $-1$?
- Defining a Galois Field based on primitive element versus polynomial?
- Is computer science a branch of mathematics?
- Can't find the relationship between two columns of numbers. Please Help
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- A community project: prove (or disprove) that $\sum_{n\geq 1}\frac{\sin(2^n)}{n}$ is convergent
- Alternative way of expressing a quantied statement with "Some"
Popular # Hahtags
real-analysis
calculus
linear-algebra
probability
abstract-algebra
integration
sequences-and-series
combinatorics
general-topology
matrices
functional-analysis
complex-analysis
geometry
group-theory
algebra-precalculus
probability-theory
ordinary-differential-equations
limits
analysis
number-theory
measure-theory
elementary-number-theory
statistics
multivariable-calculus
functions
derivatives
discrete-mathematics
differential-geometry
inequality
trigonometry
Popular Questions
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- How to find mean and median from histogram
- Difference between "≈", "≃", and "≅"
- Easy way of memorizing values of sine, cosine, and tangent
- How to calculate the intersection of two planes?
- What does "∈" mean?
- If you roll a fair six sided die twice, what's the probability that you get the same number both times?
- Probability of getting exactly 2 heads in 3 coins tossed with order not important?
- Fourier transform for dummies
- Limit of $(1+ x/n)^n$ when $n$ tends to infinity