Operator theory

• How to Find the Eigenfunctions of a Differential Operator
  Find eigenfunctions by solving the operator’s eigenvalue ODE, then applying its domain and boundary conditions to identify allowed eigenvalues.
• Eigenvalues of the Left-Shift Plus Right-Shift Operator
  For the unilateral shifts on ℓ² of the nonnegative integers, their sum has spectrum [−2, 2] but no eigenvalues.
• How Do Eigenvectors of Tensor Product Operators Work?
  If Ax = λx and By = μy, then (A ⊗ B)(x ⊗ y) = λμ(x ⊗ y); degeneracies can also yield sums.
• What Is a Core for an Unbounded Operator?
  A core is a subdomain dense in an unbounded operator's graph norm; learn the precise criterion and see a multiplication-operator example.
• Why an Infinite-Rank Compact Operator Has Nonclosed Range
  A compact operator between Banach spaces with closed range must have finite rank. The quotient-space proof explains why infinite rank forces nonclosed range.
• What Does the Polarization Identity Imply for Operators?
  The polarization identity shows that a linear norm-preserving map on a Hilbert space preserves inner products and satisfies T* T = I.
• What Is the Adjoint of an Integral Operator on $L^p$?
  For an integral operator on Lp, the adjoint reverses the kernel variables; the dual space and any complex conjugation depend on p and convention.