Hilbert spaces

• Why Does an Isometric Linear Operator Satisfy $T^*T=I$?
  A linear isometry preserves inner products, and the adjoint identity then shows that its adjoint composed with it is the identity.
• Does an Invertible Adjoint Imply an Invertible Operator?
  If the adjoint of a bounded Hilbert-space operator is invertible, then the operator itself is invertible, with inverse equal to the adjoint inverse.
• How Do You Expand an Inner Product of Finite Sums?
  Expand an inner product of finite sums by bilinearity or sesquilinearity, then simplify cross terms carefully, especially when calculating a squared norm.
• Functional Analysis for Quantum Mechanics: What You Need to Know
  Functional analysis supplies quantum mechanics with Hilbert spaces, self-adjoint operators, spectra, and the rigorous treatment of measurement and evolution.
• 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.
• 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.