An operator T on a Hilbert space H is self-adjoint if:

Self-Adjoint Operators Quiz

Quiz
•
Mathematics
•
University
•
Medium
Mahalakshmi Murugan
Used 1+ times
FREE Resource
20 questions
Show all answers
1.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
T^2 = I
⟨Tx,y⟩ = ⟨x,Ty⟩ for all x,y ∈ H
T = T^{-1}
∥Tx∥ = ∥x∥ for all x ∈ H
2.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
Every self-adjoint operator is:
Nonlinear
Bounded (if defined on entire space)
Unbounded
Compact
3.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
The spectrum of a bounded self-adjoint operator lies in:
The complex plane
The unit circle
The real line
The imaginary axis
4.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
Which of the following is true for a self-adjoint operator T?
T^* = -T
T has no eigenvalues
T = T^*
T^*T = TT^*
5.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
The set of all self-adjoint operators on a Hilbert space forms:
A group
A real vector space
A ring
A normed space only
6.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
If T is self-adjoint, then for every eigenvalue λ:
λ is purely imaginary
λ is real
λ is complex
λ = 0
7.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
A self-adjoint operator is always:
Compact
Diagonalizable (in separable Hilbert space)
Finite-rank
Non-invertible
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