Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound

Let $\mathbb{R}$ be the ordered field of real numbers and $\mathbb{N}$ the set of natural numbers; sequences are indexed by $\mathbb{N}$ as in Sequence in a Set, and a real sequence converges to $0$ in the sense of Limit of a Sequence of Real Numbers. Let $\mathbb{C}\supseteq\mathbb{R}$ be the field of complex numbers, with imaginary unit $i$, conjugation $z\mapsto\overline{z}$, modulus $|z|$ and real and imaginary parts $\operatorname{Re}z$, $\operatorname{Im}z$. Let $V$, $W$ and $U$ be complex inner product spaces, with inner products $\langle\cdot,\cdot\rangle_{V}$, $\langle\cdot,\cdot\rangle_{W}$, $\langle\cdot,\cdot\rangle_{U}$ and induced norms $\lVert\cdot\rVert_{V}$, $\lVert\cdot\rVert_{W}$, $\lVert\cdot\rVert_{U}$, which are norms by claim 2 of The Induced Norm is a Norm, and Induces a Metric. Bounded linear maps, bounds, $\mathcal{L}(V,W)$ and $\lVert T\rVert_{\mathrm{op}}$ are those of Bounded Linear Maps between Complex Normed Spaces and the Operator Norm, and adjoints are those of Adjoint of a Linear Map between Complex Inner Product Spaces. For linear maps $S,T:V\to W$ and $R:W\to U$ and for $c\in\mathbb{C}$, the linear maps $S+T$, $cT$ and $RT$ send $v\in V$ to $Sv+Tv$, $c\,(Tv)$ and $R(Tv)$; $I_{V}$ is the identity map of $V$. A complex Hilbert space is as in Complex Hilbert Space; convergence in a complex inner product space refers to the metric $(u,u')\mapsto\lVert u-u'\rVert$ of that definition and to convergence in a metric space.

1. (Least bound)¶ Let $T\in\mathcal{L}(V,W)$. Then $\lVert T\rVert_{\mathrm{op}}$ is a bound for $T$, and $\lVert T\rVert_{\mathrm{op}}\le C$ for every bound $C$ for $T$. A linear map $T:V\to V$ belongs to $\mathcal{L}(V)$ if and only if it is a bounded linear operator on $V$, and then $\lVert T\rVert_{\mathrm{op}}$ is its operator norm.

2. (Operations)¶ Let $S,T\in\mathcal{L}(V,W)$, $R\in\mathcal{L}(W,U)$ and $c\in\mathbb{C}$. Then $S+T,\,cT\in\mathcal{L}(V,W)$, $RT\in\mathcal{L}(V,U)$ and $I_{V}\in\mathcal{L}(V)$; with these sums and scalar multiples $\mathcal{L}(V,W)$ is a complex vector space whose zero vector is the zero map $0$; composition distributes over sums and commutes with scalar multiples; and

3. (Underlying real structure)¶ Let $X$ be a complex vector space and let $h:X\times X\to\mathbb{C}$ satisfy, for all $u,v,w\in X$ and $c\in\mathbb{C}$,

with $h(v,v)$ a real number and $0\le h(v,v)$. Then $X$, with its addition and with the multiplication by the real scalars, is a vector space over $\mathbb{R}$; the map $\beta(u,v)=\operatorname{Re}h(u,v)$ is symmetric, bilinear over $\mathbb{R}$ and positive semidefinite in the sense of Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences; and for all $u,v\in X$

For $X=V$ and $h=\langle\cdot,\cdot\rangle_{V}$ the map $\beta$ is an inner product on this real vector space, written $\langle\cdot,\cdot\rangle_{V,\mathbb{R}}$, whose norm is $\lVert\cdot\rVert_{V}$; if $V$ is a complex Hilbert space, then $V$ with $\langle\cdot,\cdot\rangle_{V,\mathbb{R}}$ is a real Hilbert space.

4. (Uniqueness of adjoints)¶ A linear map $T:V\to W$ has at most one adjoint; when it has one, it is written $T^{*}$.

5. (Existence of adjoints)¶ If $V$ is a complex Hilbert space and $T\in\mathcal{L}(V,W)$, then $T$ has an adjoint, $T^{*}\in\mathcal{L}(W,V)$, and $\lVert T^{*}\rVert_{\mathrm{op}}=\lVert T\rVert_{\mathrm{op}}$.

6. (Calculus of adjoints)¶ Let $S,T:V\to W$ and $R:W\to U$ be linear maps that have adjoints, and let $c\in\mathbb{C}$. Then $T$ is the adjoint of $T^{*}$; the maps $S+T$, $cT$ and $RT$ have the adjoints $S^{*}+T^{*}$, $\overline{c}\,T^{*}$ and $T^{*}R^{*}$; and $I_{V}$ is its own adjoint. A linear map $A:V\to V$ is self-adjoint if and only if $A$ is an adjoint of $A$. Moreover $\langle v,T^{*}Tv\rangle_{V}=\lVert Tv\rVert_{W}^{2}$ for every $v\in V$, and $T^{*}T$ is self-adjoint and positive semi-definite.

7. (Completeness)¶ Let $W$ be a complex Hilbert space and let $(T_{k})_{k\in\mathbb{N}}$ be a sequence in $\mathcal{L}(V,W)$ such that for every real $\varepsilon>0$ there is $N\in\mathbb{N}$ with $\lVert T_{k}-T_{l}\rVert_{\mathrm{op}}<\varepsilon$ for all $k,l\ge N$, where $T_{k}-T_{l}=T_{k}+(-1)T_{l}$. Then there is $T\in\mathcal{L}(V,W)$ such that the real sequence $\bigl(\lVert T_{k}-T\rVert_{\mathrm{op}}\bigr)_{k\in\mathbb{N}}$ converges to $0$; for every such $T$ and every $v\in V$, the sequence $(T_{k}v)_{k\in\mathbb{N}}$ converges to $Tv$ in $W$.

8. (Quadratic-form bound)¶ Let $A:V\to V$ be a self-adjoint linear map and let $c\ge0$ be real with $|\langle v,Av\rangle_{V}|\le c\,\lVert v\rVert_{V}^{2}$ for every $v\in V$. Then $A\in\mathcal{L}(V)$ and $\lVert A\rVert_{\mathrm{op}}\le c$.