Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation

1. (Numbers)¶ The conventions of The Real Numbers: Standing Notation and Background are in force. $\mathbb{C}\supseteq\mathbb{R}$ is the field of complex numbers, with imaginary unit $i$, conjugation $z\mapsto\overline{z}$, modulus $|z|$ and real and imaginary parts $\operatorname{Re}z$, $\operatorname{Im}z$.

2. (Spaces)¶ The letters $V,W,U$ denote complex inner product spaces and $H,K$ denote complex Hilbert spaces. The inner product $\langle\cdot,\cdot\rangle$ is linear in its second argument and conjugate-linear in its first; $\lVert\cdot\rVert$ is the induced norm; subscripts, as in $\langle\cdot,\cdot\rangle_{H}$ and $\lVert\cdot\rVert_{H}$, name the space when several are in play; $0$ is the zero vector. A sequence is convergent or Cauchy, and a subset is dense or closed, with respect to the metric $(u,u')\mapsto\lVert u-u'\rVert$ of Complex Hilbert Space, in the sense of convergence, the Cauchy condition, density and closedness for the topology $\mathcal{T}_{d}$ of the subsets that are open in the metric space, which is a topology by Metric Open Sets Form a Topology. $\langle u,v\rangle_{V,\mathbb{R}}=\operatorname{Re}\langle u,v\rangle_{V}$ is the underlying real inner product.

3. (Maps)¶ $\mathcal{L}(V,W)$ and $\mathcal{L}(V)$ are the sets of bounded linear maps, and $\lVert T\rVert_{\mathrm{op}}$ is the operator norm. Sums $S+T$, scalar multiples $cT$ and composites $RT$ of linear maps are formed pointwise and by composition, $S-T=S+(-1)T$, and $I_{V}$ (or $I$) is the identity map of $V$. $T^{*}$ is the adjoint of $T$ when it exists, as in Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-unique. Self-adjoint and positive semi-definite linear maps $V\to V$ are those of Self-Adjoint Operator and Positive Semi-Definite Operator, and $A\ge0$ means that $A\in\mathcal{L}(V)$ is self-adjoint and positive semi-definite. For a sequence $(T_{k})$ and $T$ in $\mathcal{L}(V,W)$, "$T_{k}\to T$ in operator norm" abbreviates "the real sequence $\bigl(\lVert T_{k}-T\rVert_{\mathrm{op}}\bigr)$ converges to $0$", and for $S,T\in\mathcal{L}(V)$, "$S$ and $T$ commute" abbreviates $ST=TS$.

4. (Background)¶ The following results are in force and may be used without restating them: Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound, Elementary Properties of a Complex Inner Product, Cauchy-Schwarz Inequality in a Complex Inner Product Space, The Induced Norm is a Norm, and Induces a Metric, Properties of Complex Conjugation and Modulus, Canonical Form and Arithmetic of Complex Numbers and The Complex Numbers are Complete in the Modulus Metric.