Self-Adjoint Operators Quiz

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

• T^2 = I

• ⟨Tx,y⟩ = ⟨x,Ty⟩ for all x,y ∈ H

• T = T^{-1}

• ∥Tx∥ = ∥x∥ for all x ∈ H

Every self-adjoint operator is:

• Nonlinear

• Bounded (if defined on entire space)

• Unbounded

• Compact

The spectrum of a bounded self-adjoint operator lies in:

• The complex plane

• The unit circle

• The real line

• The imaginary axis

Which of the following is true for a self-adjoint operator T?

• T^* = -T

• T has no eigenvalues

• T = T^*

• T^*T = TT^*

The set of all self-adjoint operators on a Hilbert space forms:

• A group

• A real vector space

• A ring

• A normed space only

If T is self-adjoint, then for every eigenvalue λ:

• λ is purely imaginary

• λ is real

• λ is complex

• λ = 0

A self-adjoint operator is always:

• Compact

• Diagonalizable (in separable Hilbert space)

• Finite-rank

• Non-invertible

In finite-dimensional Hilbert space, a self-adjoint operator corresponds to:

• A skew-symmetric matrix

• A Hermitian matrix

• A unitary matrix

• A lower triangular matrix

Which operator is always self-adjoint?

• The right shift operator

• i

• The identity operator I

• The differentiation operator on L^2

Self-adjoint operators have:

• Non-zero kernel

• Real spectrum

• Imaginary spectrum

• No eigenvectors