Find the norm of the operator on ℬ(ℋ) where ℋ=ℂ^2 which is given by the matrix [ a b

step 1 Okay, the key idea for this problem is if you want to find the standard matrix of a composition

Find the norm of the operator on ℬ(ℋ) where ℋ=ℂ^2 which is...

Find the norm of the operator on $\mathscr{B}(\mathscr{H})$ where $\mathscr{H}=\mathbb{C}^{2}$ which is given by the matrix
$$
\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]
$$
Give your answer in terms of the numbers $S=|a|^{2}+|b|^{2}+|c|^{2}+|d|^{2}$ and $D=$ $a d-b c$.

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Key Concepts

Operator Norm

The operator norm of a bounded linear operator on a Hilbert space is defined as the supremum of the norm of the operator applied to a unit vector. In finite-dimensional spaces, this norm is equivalent to the largest singular value of the operator, representing its maximum stretching factor.

Singular Values

Singular values of a matrix are the square roots of the eigenvalues of the product of the matrix with its adjoint. They quantify the magnitude by which the matrix scales the input vectors. For a 2×2 matrix, the largest singular value is exactly the operator norm.

Matrix Invariants

Matrix invariants such as the sum of squares of the entries and the determinant play a crucial role in simplifying expressions for norms. In the given context, the sum S represents the squared Frobenius (or Hilbert-Schmidt) norm, and the determinant D captures key geometric properties. These invariants allow the largest singular value (and hence the operator norm) to be expressed in closed form.

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