Compact Operators on a Hilbert Space

Click For Summary
SUMMARY

The discussion centers on the equivalence of compact operators on a Hilbert space, specifically addressing the formula \(\mathcal{K} (\mathcal{H}) \approx \mathcal{K} (\mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H}) \approx M_{4} (\mathcal{K} (\mathcal{H}))\). Participants confirm the validity of this statement under the assumption that \(H\) is infinite-dimensional and separable. The conversation emphasizes the implications of this equivalence for understanding the structure of compact operators and their representations.

PREREQUISITES
  • Understanding of Hilbert spaces and their properties
  • Familiarity with compact operators in functional analysis
  • Knowledge of direct sums of vector spaces
  • Basic concepts of operator theory and matrix representations
NEXT STEPS
  • Study the properties of compact operators in Hilbert spaces
  • Explore the implications of the direct sum in operator theory
  • Research the structure of matrix representations of compact operators
  • Learn about separable and non-separable Hilbert spaces
USEFUL FOR

Mathematicians, students of functional analysis, and anyone interested in the theoretical aspects of compact operators on Hilbert spaces.

lunde
Messages
13
Reaction score
0
Hello, I hope I am asking this in the right area of the forums. I wanted to ask if the following formula is true (assuming H is infinite dimensional and separable):

[tex]\mathcal{K} (\mathcal{H}) \approx \mathcal{K} (\mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H})\approx M_{4} (\mathcal{K} (\mathcal{H}))[/tex]

I'm pretty sure this is true, but I am worried I am crazy, because I don't understand how every compact operator could secretly be 16 compact operators
 
Last edited:
Physics news on Phys.org
Please only post a question once.
 

Similar threads

Graduate Operator mapping in Hilbert space
  • · Replies 35 ·
2
Replies
35
Views
4K
Graduate How to derive the quantum detailed balance condition?
  • · Replies 0 ·
Replies
0
Views
1K
Graduate Equivalence Relation to define the tensor product of Hilbert spaces
  • · Replies 1 ·
Replies
1
Views
2K
Homework Help: Quantum Information Theory 1
  • · Replies 0 ·
Replies
0
Views
2K
Undergrad Are all (pure) states physically realizable?
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad Quantum particle's state in momentum eigenfunctions basis
  • · Replies 16 ·
Replies
16
Views
2K
Insights Hilbert Spaces And Their Relatives - Operators
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Action of an Hermitian unitary operator member of SU(2)
  • · Replies 15 ·
Replies
15
Views
1K
Graduate Link between Kraus operators and PVMs
  • · Replies 0 ·
Replies
0
Views
1K
Undergrad Understanding Hilbert Vector Spaces
  • · Replies 3 ·
Replies
3
Views
2K