by William L. Paschke
edition 0.9
image/svg+xml
Frontmatter
I. Hilbert Space
II. Bounded Operators
III. Compact Operators
IV. The Spectral Theorem
Index and References
Let HHilbert space\( H \) be a Hilbert space and let φfunctional∈element ofHHilbert space* \( φ\in {H}^{*} \). Then there exists a unique yvector∈element ofHHilbert space \( y\in H \) such that φfunctional(xvector)=equals〈xvector, yvector〉 \( φ\mathopen{}\left( x\right)\mathclose{}= \mathopen{}\left\langle{}x, y\right\rangle\mathclose{} \) for all xvector∈element ofHHilbert space \( x\in H \). Further, ‖φfunctional‖=equals‖yvector‖ \( \mathopen{}\left\lVert{}φ\right\rVert\mathclose{}= \mathopen{}\left\lVert{}y\right\rVert\mathclose{} \).
Proof. We may assume φfunctional≠not equal to0zero \( φ\neq 0 \). Since φfunctional\( φ \) is continuous, φfunctional−1inverse(0zero) \( {φ}^{-1}\mathopen{}\left( 0\right)\mathclose{} \) is a closed proper subspace of HHilbert space\( H \). So, φfunctional−1inverse(0zero) ⊥ \( { {φ}^{-1}\mathopen{}\left( 0\right)\mathclose{} }^{\perp} \) is non-trivial (from Proposition I.61) and there is a non-zero vector yvector0zero∈element of φfunctional−1inverse(0zero) ⊥ \( {y}_{0}\in { {φ}^{-1}\mathopen{}\left( 0\right)\mathclose{} }^{\perp} \). Thus φfunctional(yvector0zero)≠not equal to0zero \( φ\mathopen{}\left( {y}_{0}\right)\mathclose{}\neq 0 \) (else 〈yvector0zero, yvector0zero〉=equals0zero \( \mathopen{}\left\langle{}{y}_{0}, {y}_{0}\right\rangle\mathclose{}= 0 \)). For xvector∈element ofHHilbert space \( x\in H \) consider xvector-minus φfunctional(xvector) φfunctional(yvector0zero) timesyvector0zero \( x-\frac{φ\mathopen{}\left( x\right)\mathclose{}}{φ\mathopen{}\left( {y}_{0}\right)\mathclose{}}{y}_{0} \), which belongs to φfunctional−1inverse(0zero) \( {φ}^{-1}\mathopen{}\left( 0\right)\mathclose{} \). Since yvector0zero⊥φfunctional−1inverse(0zero) \( {y}_{0}\perp {φ}^{-1}\mathopen{}\left( 0\right)\mathclose{} \), we get that 〈xvector-minus φfunctional(xvector) φfunctional(yvector0zero) timesyvector0zero, yvector0zero〉=equals0zero \[ \mathopen{}\left\langle{}x-\frac{φ\mathopen{}\left( x\right)\mathclose{}}{φ\mathopen{}\left( {y}_{0}\right)\mathclose{}}{y}_{0}, {y}_{0}\right\rangle\mathclose{}= 0 \] and 〈xvector, yvector0zero〉=equals φfunctional(xvector) φfunctional(yvector0zero) times〈yvector0zero, yvector0zero〉 ; \[ \mathopen{}\left\langle{}x, {y}_{0}\right\rangle\mathclose{}= \frac{φ\mathopen{}\left( x\right)\mathclose{}}{φ\mathopen{}\left( {y}_{0}\right)\mathclose{}}\mathopen{}\left\langle{}{y}_{0}, {y}_{0}\right\rangle\mathclose{} \text{;} \] that is, φfunctional(xvector)=equals φfunctional(yvector0zero) 〈yvector0zero, yvector0zero〉 times〈xvector, yvector0zero〉 \( φ\mathopen{}\left( x\right)\mathclose{}= \frac{φ\mathopen{}\left( {y}_{0}\right)\mathclose{}}{\mathopen{}\left\langle{}{y}_{0}, {y}_{0}\right\rangle\mathclose{}}\mathopen{}\left\langle{}x, {y}_{0}\right\rangle\mathclose{} \). So, yvector=equals φfunctional(yvector0zero) ¯ 〈yvector0zero, yvector0zero〉 timesyvector0zero \( y= \frac{\overline{ φ\mathopen{}\left( {y}_{0}\right)\mathclose{} }}{\mathopen{}\left\langle{}{y}_{0}, {y}_{0}\right\rangle\mathclose{}}{y}_{0} \) works.
For uniqueness: if also φfunctional(xvector)=equals〈xvector, z〉 \( φ\mathopen{}\left( x\right)\mathclose{}= \mathopen{}\left\langle{}x, z\right\rangle\mathclose{} \), then 〈xvector, yvector-minusz〉=equals0zero \( \mathopen{}\left\langle{}x, y-z\right\rangle\mathclose{}= 0 \) for all xvector∈element ofHHilbert space \( x\in H \); in particular, 〈yvector-minusz, yvector-minusz〉=equals0zero \( \mathopen{}\left\langle{}y-z, y-z\right\rangle\mathclose{}= 0 \).
To show ‖φfunctional‖=equals‖yvector‖ \( \mathopen{}\left\lVert{}φ\right\rVert\mathclose{}= \mathopen{}\left\lVert{}y\right\rVert\mathclose{} \), we have |modulusφfunctional(xvector)|modulus=equals|modulus〈xvector, yvector〉|modulus≤less than or equal to‖xvector‖times‖yvector‖ \( \mathopen{}\left\lvert{}φ\mathopen{}\left( x\right)\mathclose{}\right\rvert\mathclose{}= \mathopen{}\left\lvert{}\mathopen{}\left\langle{}x, y\right\rangle\mathclose{}\right\rvert\mathclose{}\leq \mathopen{}\left\lVert{}x\right\rVert\mathclose{}\mathopen{}\left\lVert{}y\right\rVert\mathclose{} \). So, ‖yvector‖≥greater than or equal to‖φfunctional‖ \( \mathopen{}\left\lVert{}y\right\rVert\mathclose{}\geq \mathopen{}\left\lVert{}φ\right\rVert\mathclose{} \). Also, |modulusφfunctional(yvector)|modulus=equals|modulus〈yvector, yvector〉|modulus=equals‖yvector‖times‖yvector‖ \( \mathopen{}\left\lvert{}φ\mathopen{}\left( y\right)\mathclose{}\right\rvert\mathclose{}= \mathopen{}\left\lvert{}\mathopen{}\left\langle{}y, y\right\rangle\mathclose{}\right\rvert\mathclose{}= \mathopen{}\left\lVert{}y\right\rVert\mathclose{}\mathopen{}\left\lVert{}y\right\rVert\mathclose{} \). So, |modulusφfunctional(yvector‖yvector‖)|modulus=equals‖yvector‖ \( \mathopen{}\left\lvert{}φ\mathopen{}\left( \frac{y}{\mathopen{}\left\lVert{}y\right\rVert\mathclose{}}\right)\mathclose{}\right\rvert\mathclose{}= \mathopen{}\left\lVert{}y\right\rVert\mathclose{} \). Thus ‖φfunctional‖≥greater than or equal to‖yvector‖ \( \mathopen{}\left\lVert{}φ\right\rVert\mathclose{}\geq \mathopen{}\left\lVert{}y\right\rVert\mathclose{} \).
Let HHilbert space\( H \) and JHilbert space\( J \) be Hilbert spaces, and let Tlinear map∈element ofℒbounded linear operators(HHilbert spaceJHilbert space) \( T\in \mathcal{L}\mathopen{}\left( H, J\right)\mathclose{} \). Then there exists Tlinear map*adjoint∈element ofℒbounded linear operators(JHilbert spaceHHilbert space) \( T^{*}\in \mathcal{L}\mathopen{}\left( J, H\right)\mathclose{} \) such that 〈Tlinear map(xvector), yvector〉JHilbert space=equals〈xvector, Tlinear map*(yvector)〉HHilbert space \( \mathopen{}\left\langle{}T\mathopen{}\left( x\right)\mathclose{}, y\right\rangle\mathclose{}_{J}= \mathopen{}\left\langle{}x, T^{*}\mathopen{}\left( y\right)\mathclose{}\right\rangle\mathclose{}_{H} \) for all xvector∈element ofHHilbert space \( x\in H \), yvector∈element ofJHilbert space \( y\in J \). Further, Tlinear map*\( T^{*} \) is unique, (Tlinear map*)*=equalsTlinear map \( \mathopen{}\left( T^{*}\right)\mathclose{}^{*}= T \), and ‖Tlinear map*‖=equals‖Tlinear map‖ \( \mathopen{}\left\lVert{} T^{*}\right\rVert\mathclose{}= \mathopen{}\left\lVert{}T\right\rVert\mathclose{} \). Tlinear map*\( T^{*} \) is called the adjoint of Tlinear map\( T \).
Proof. Fix yvector∈element ofJHilbert space \( y\in J \) and consider φfunctional:mapsHHilbert space→toCcomplex numbers \( φ : H \to \mathbb{C} \) defined by φfunctional(xvector)=equals〈Tlinear map(xvector), yvector〉 \( φ\mathopen{}\left( x\right)\mathclose{}= \mathopen{}\left\langle{}T\mathopen{}\left( x\right)\mathclose{}, y\right\rangle\mathclose{} \). We have |modulusφfunctional(xvector)|modulus≤less than or equal to‖Tlinear map(xvector)‖times‖yvector‖≤less than or equal to‖Tlinear map‖times‖xvector‖times‖yvector‖=equals(‖Tlinear map‖times‖yvector‖)times‖xvector‖ \( \mathopen{}\left\lvert{}φ\mathopen{}\left( x\right)\mathclose{}\right\rvert\mathclose{}\leq \mathopen{}\left\lVert{}T\mathopen{}\left( x\right)\mathclose{}\right\rVert\mathclose{}\mathopen{}\left\lVert{}y\right\rVert\mathclose{}\leq \mathopen{}\left\lVert{}T\right\rVert\mathclose{}\mathopen{}\left\lVert{}x\right\rVert\mathclose{}\mathopen{}\left\lVert{}y\right\rVert\mathclose{}= \mathopen{}\left(\mathopen{}\left\lVert{}T\right\rVert\mathclose{}\mathopen{}\left\lVert{}y\right\rVert\mathclose{}\right)\mathclose{}\mathopen{}\left\lVert{}x\right\rVert\mathclose{} \). So, φfunctional∈element ofHHilbert space* \( φ\in H^{*} \) (with ‖φfunctional‖≤less than or equal to‖Tlinear map‖times‖yvector‖ \( \mathopen{}\left\lVert{}φ\right\rVert\mathclose{}\leq \mathopen{}\left\lVert{}T\right\rVert\mathclose{}\mathopen{}\left\lVert{}y\right\rVert\mathclose{} \)). Thus we get an implementing vector, call it Tlinear map*(yvector) \( T^{*}\mathopen{}\left( y\right)\mathclose{} \), such that φfunctional(xvector)=equals〈xvector, Tlinear map*(yvector)〉 \( φ\mathopen{}\left( x\right)\mathclose{}= \mathopen{}\left\langle{}x, T^{*}\mathopen{}\left( y\right)\mathclose{}\right\rangle\mathclose{} \) for all xvector\( x \). Easily, Tlinear map*:mapsJHilbert space→toHHilbert space \( T^{*} : J \to H \) is linear and ‖Tlinear map*(yvector)‖=equals‖φfunctional‖≤less than or equal to‖Tlinear map‖times‖yvector‖ \( \mathopen{}\left\lVert{} T^{*}\mathopen{}\left( y\right)\mathclose{}\right\rVert\mathclose{}= \mathopen{}\left\lVert{}φ\right\rVert\mathclose{}\leq \mathopen{}\left\lVert{}T\right\rVert\mathclose{}\mathopen{}\left\lVert{}y\right\rVert\mathclose{} \). So, Tlinear map*∈element ofℒbounded linear operators(JHilbert spaceHHilbert space) \( T^{*}\in \mathcal{L}\mathopen{}\left( J, H\right)\mathclose{} \) with ‖Tlinear map*‖≤less than or equal to‖Tlinear map‖ \( \mathopen{}\left\lVert{} T^{*}\right\rVert\mathclose{}\leq \mathopen{}\left\lVert{}T\right\rVert\mathclose{} \). At this point it is clear that (Tlinear map*)*=equalsTlinear map \( \mathopen{}\left( T^{*}\right)\mathclose{}^{*}= T \), so we have ‖Tlinear map‖=equals‖(Tlinear map*)*‖≤less than or equal to‖Tlinear map*‖ \( \mathopen{}\left\lVert{}T\right\rVert\mathclose{}= \mathopen{}\left\lVert{} \mathopen{}\left( T^{*}\right)\mathclose{}^{*}\right\rVert\mathclose{}\leq \mathopen{}\left\lVert{} T^{*}\right\rVert\mathclose{} \). Thus ‖Tlinear map*‖=equals‖Tlinear map‖ \( \mathopen{}\left\lVert{} T^{*}\right\rVert\mathclose{}= \mathopen{}\left\lVert{}T\right\rVert\mathclose{} \).
implementing vector
The adjoint of a map from Ccomplex numbersninteger\( {\mathbb{C}}^{n} \) to Ccomplex numbersminteger\( {\mathbb{C}}^{m} \) is obtained by taking the conjugate transpose of the representing matrix: Tlinear map=equals(sequence treal numberiintegerjinteger )sequence \( T= \mathopen{}\left( {t}_{ij} \right)\mathclose{} \) to Tlinear map*=equals(sequence treal numberjintegeriinteger ¯complex conjugate )sequence \( T^{*}= \mathopen{}\left( \overline{ {t}_{ji} } \right)\mathclose{} \).
If Tlinear map\( T \) is multiplication by a bounded measurable φbounded measurable function\( φ \) on L2Lebesgue space(Ωmeasure space) \( \mathrm{L}^{\mathrm{2}}\mathopen{}\left( Ω\right)\mathclose{} \), then Tlinear map*\( T^{*} \) is multiplication by φbounded measurable function¯complex conjugate\( \overline{φ} \) because 〈Tlinear map*(ffunction), ggroup element〉=equals〈ffunction, Tlinear map(ggroup element)〉=equals∫integralΩmeasure spaceffunctiontimesTlinear map(ggroup element)¯complex conjugate=equals∫integralΩmeasure spaceffunctiontimesφbounded measurable functiontimesggroup element¯complex conjugate=equals∫integralΩmeasure space (φbounded measurable function¯complex conjugatetimesffunction)timesggroup element¯complex conjugate =equals〈φbounded measurable function¯complex conjugate(ffunction), ggroup element〉 , \[ \mathopen{}\left\langle{} T^{*}\mathopen{}\left( f\right)\mathclose{}, g\right\rangle\mathclose{}= \mathopen{}\left\langle{}f, T\mathopen{}\left( g\right)\mathclose{}\right\rangle\mathclose{}= \int _{Ω}{}f\overline{T\mathopen{}\left( g\right)\mathclose{}}= \int _{Ω}{}f\overline{φg}= \int _{Ω}{} \mathopen{}\left(\overline{φ}f\right)\mathclose{}\overline{g} = \mathopen{}\left\langle{}\overline{φ}\mathopen{}\left( f\right)\mathclose{}, g\right\rangle\mathclose{} \text{,} \] i.e. Tlinear map*(ffunction)=equalsφbounded measurable function¯complex conjugate(ffunction) \( T^{*}\mathopen{}\left( f\right)\mathclose{}= \overline{φ}\mathopen{}\left( f\right)\mathclose{} \).
Let Tlinear map\( T \) be the Volterra operator on L2Lebesgue space([interval0zero, 1one]interval) \( \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{} \): (Tlinear map(ffunction))(xreal number)=equals∫integral0zeroxreal numberffunction(treal number)dtreal number \( \mathopen{}\left(T\mathopen{}\left( f\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}= \int _{0}^{x}{}f\mathopen{}\left( t\right)\mathclose{}\,\mathrm{d}t \). Then 〈Tlinear map*(ffunction), ggroup element〉=equals〈ffunction, Tlinear map(ggroup element)〉=equals∫integral0zero1one ffunction(xreal number)timesTlinear map(ggroup element(xreal number))¯complex conjugate dxreal number=equals∫integral0zero1one ffunction(xreal number)times(∫integral0zeroxreal numberggroup element(treal number)¯complex conjugatedtreal number) dxreal number=equals∫integral0zero1one ∫integral0zeroxreal number ffunction(xreal number)timesggroup element(treal number)¯complex conjugate dtreal number dxreal number=equals∫integral0zero1one ∫integraltreal number1one ffunction(xreal number)timesggroup element(treal number)¯complex conjugate dxreal number dtreal number=equals∫integral0zero1one ∫integral0zero1one kpiecewise function(xreal numbertreal number)timesffunction(xreal number)timesggroup element(treal number)¯complex conjugate dxreal number dtreal number , \[ \mathopen{}\left\langle{} T^{*}\mathopen{}\left( f\right)\mathclose{}, g\right\rangle\mathclose{}= \mathopen{}\left\langle{}f, T\mathopen{}\left( g\right)\mathclose{}\right\rangle\mathclose{}= \int _{0}^{1}{} f\mathopen{}\left( x\right)\mathclose{}\overline{T\mathopen{}\left( g\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}} \,\mathrm{d}x= \int _{0}^{1}{} f\mathopen{}\left( x\right)\mathclose{}\mathopen{}\left(\int _{0}^{x}{}\overline{g\mathopen{}\left( t\right)\mathclose{}}\,\mathrm{d}t\right)\mathclose{} \,\mathrm{d}x= \int _{0}^{1}{} \int _{0}^{x}{} f\mathopen{}\left( x\right)\mathclose{}\overline{g\mathopen{}\left( t\right)\mathclose{}} \,\mathrm{d}t \,\mathrm{d}x= \int _{0}^{1}{} \int _{t}^{1}{} f\mathopen{}\left( x\right)\mathclose{}\overline{g\mathopen{}\left( t\right)\mathclose{}} \,\mathrm{d}x \,\mathrm{d}t= \int _{0}^{1}{} \int _{0}^{1}{} k\mathopen{}\left( x, t\right)\mathclose{}f\mathopen{}\left( x\right)\mathclose{}\overline{g\mathopen{}\left( t\right)\mathclose{}} \,\mathrm{d}x \,\mathrm{d}t \text{,} \] where kpiecewise function(xreal numbertreal number)=equals{cases1one, treal number≤less than or equal toxreal number; 0zero, otherwise.} \[ k\mathopen{}\left( x, t\right)\mathclose{}= \begin{cases}1, & t\leq x; \\ 0, & \text{otherwise.}\end{cases} \] By Fubini's theorem, we get ∫integral0zero1one ∫integral0zero1one kpiecewise function(xreal numbertreal number)timesffunction(xreal number)timesggroup element(treal number)¯complex conjugate dxreal number dtreal number=equals∫integral0zero1one (∫integraltreal number1oneffunction(xreal number)dxreal number)timesggroup element(treal number)¯complex conjugate dtreal number=equals〈Tlinear map*(ffunction), ggroup element〉 , \[ \int _{0}^{1}{} \int _{0}^{1}{} k\mathopen{}\left( x, t\right)\mathclose{}f\mathopen{}\left( x\right)\mathclose{}\overline{g\mathopen{}\left( t\right)\mathclose{}} \,\mathrm{d}x \,\mathrm{d}t= \int _{0}^{1}{} \mathopen{}\left(\int _{t}^{1}{}f\mathopen{}\left( x\right)\mathclose{}\,\mathrm{d}x\right)\mathclose{}\overline{g\mathopen{}\left( t\right)\mathclose{}} \,\mathrm{d}t= \mathopen{}\left\langle{} T^{*}\mathopen{}\left( f\right)\mathclose{}, g\right\rangle\mathclose{} \text{,} \] so that (Tlinear map*(ffunction))(xreal number)=equals∫integralxreal number1oneffunction(treal number)dtreal number \( \mathopen{}\left( T^{*}\mathopen{}\left( f\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}= \int _{x}^{1}{}f\mathopen{}\left( t\right)\mathclose{}\,\mathrm{d}t \). Notice that for this operator (Tlinear map+plusTlinear map*)(ffunction) \( \mathopen{}\left(T+ T^{*}\right)\mathclose{}\mathopen{}\left( f\right)\mathclose{} \) is a constant function. That is, (Tlinear map+plusTlinear map*)(ffunction)=equals〈ffunction, 1constant function〉times1constant function \( \mathopen{}\left(T+ T^{*}\right)\mathclose{}\mathopen{}\left( f\right)\mathclose{}= \mathopen{}\left\langle{}f, \mathbb{1}\right\rangle\mathclose{}\mathbb{1} \), which is the projection of ffunction\( f \) on Ccomplex numberstimes1constant function \( \mathbb{C}\mathbb{1} \).
Recall the sets Eedge set\( E \) and Vvertex set\( V \) and maps iinitial point map,tterminal point map:mapsEedge set→toVvertex set \( i\text{,}t : E \to V \) with supsupremumvvertex∈element ofVvertex set (#cardinality(iinitial point map−1inverse(vvertex))+plus#cardinality(tterminal point map−1inverse(vvertex))) <less than∞infinity . \[ \sup_{v\in V}{} \mathopen{}\left({\#}\mathopen{}\left({i}^{-1}\mathopen{}\left( v\right)\mathclose{}\right)\mathclose{}+{\#}\mathopen{}\left({t}^{-1}\mathopen{}\left( v\right)\mathclose{}\right)\mathclose{}\right)\mathclose{} \lt \infty \text{.} \] Get Tlinear map:mapsl2(Vvertex set)→tol2(Eedge set) \( T : \mathrm{l}^{0}\mathopen{}\left( V\right)\mathclose{} \to \mathrm{l}^{0}\mathopen{}\left( E\right)\mathclose{} \) defined by (Tlinear map(ffunction))(eedge)=equalsffunction(iinitial point map(eedge))-minusffunction(tterminal point map(eedge)) \( \mathopen{}\left(T\mathopen{}\left( f\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( e\right)\mathclose{}= f\mathopen{}\left( i\mathopen{}\left( e\right)\mathclose{}\right)\mathclose{}-f\mathopen{}\left( t\mathopen{}\left( e\right)\mathclose{}\right)\mathclose{} \). What is the adjoint Tlinear map*:mapsl2(Eedge set)→tol2(Vvertex set) \( T^{*} : \mathrm{l}^{0}\mathopen{}\left( E\right)\mathclose{} \to \mathrm{l}^{0}\mathopen{}\left( V\right)\mathclose{} \)? For γfunction∈element ofl2(Eedge set) \( γ\in \mathrm{l}^{0}\mathopen{}\left( E\right)\mathclose{} \), ffunction∈element ofl2(Vvertex set) \( f\in \mathrm{l}^{0}\mathopen{}\left( V\right)\mathclose{} \), 〈Tlinear map*(γfunction), ffunction〉=equals〈γfunction, Tlinear map(ffunction)〉=equals∑summationeedge∈element ofEedge set γfunction(eedge)times (Tlinear map(ffunction))(eedge) ¯complex conjugate =equals∑summationeedge∈element ofEedge set γfunction(eedge)times(ffunction(iinitial point map(eedge))¯complex conjugate-minusffunction(tterminal point map(eedge))¯complex conjugate) =equals∑summationvvertex∈element ofVvertex setffunction(vvertex)¯complex conjugatetimes∑summation{seteedge|such thatiinitial point map(eedge)=equalsvvertex}setγfunction(eedge)-minus∑summationwvector∈element ofVvertex setffunction(wvector)¯complex conjugatetimes∑summation{seteedge¯complex conjugate|such thattterminal point map(eedge¯complex conjugate)=equalswvector}setγfunction(eedge¯complex conjugate)=equals∑summationvvertex∈element ofVvertex setffunction(vvertex)¯complex conjugatetimes(∑summation{seteedge|such thatiinitial point map(eedge)=equalsvvertex}setγfunction(eedge)-minus∑summation{seteedge¯complex conjugate|such thattterminal point map(eedge¯complex conjugate)=equalsvvertex}setγfunction(eedge¯complex conjugate)) , \[ \mathopen{}\left\langle{} T^{*}\mathopen{}\left( γ\right)\mathclose{}, f\right\rangle\mathclose{}= \mathopen{}\left\langle{}γ, T\mathopen{}\left( f\right)\mathclose{}\right\rangle\mathclose{}= \sum_{e\in E}{} γ\mathopen{}\left( e\right)\mathclose{}\overline{ \mathopen{}\left(T\mathopen{}\left( f\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( e\right)\mathclose{} } = \sum_{e\in E}{} γ\mathopen{}\left( e\right)\mathclose{}\mathopen{}\left(\overline{f\mathopen{}\left( i\mathopen{}\left( e\right)\mathclose{}\right)\mathclose{}}-\overline{f\mathopen{}\left( t\mathopen{}\left( e\right)\mathclose{}\right)\mathclose{}}\right)\mathclose{} = \sum_{v\in V}{}\overline{f\mathopen{}\left( v\right)\mathclose{}}\sum_{\mathopen{}\left\{\, e\,\middle\vert\, i\mathopen{}\left( e\right)\mathclose{}= v\,\right\}\mathclose{}}{}γ\mathopen{}\left( e\right)\mathclose{}-\sum_{w\in V}{}\overline{f\mathopen{}\left( w\right)\mathclose{}}\sum_{\mathopen{}\left\{\, \overline{e}\,\middle\vert\, t\mathopen{}\left( \overline{e}\right)\mathclose{}= w\,\right\}\mathclose{}}{}γ\mathopen{}\left( \overline{e}\right)\mathclose{}= \sum_{v\in V}{}\overline{f\mathopen{}\left( v\right)\mathclose{}}\mathopen{}\left(\sum_{\mathopen{}\left\{\, e\,\middle\vert\, i\mathopen{}\left( e\right)\mathclose{}= v\,\right\}\mathclose{}}{}γ\mathopen{}\left( e\right)\mathclose{}-\sum_{\mathopen{}\left\{\, \overline{e}\,\middle\vert\, t\mathopen{}\left( \overline{e}\right)\mathclose{}= v\,\right\}\mathclose{}}{}γ\mathopen{}\left( \overline{e}\right)\mathclose{}\right)\mathclose{} \text{,} \] whence Tlinear map*(γfunction)(vvertex)=equals∑summation{seteedge|such thatiinitial point map(eedge)=equalsvvertex}setγfunction(eedge)-minus∑summation{seteedge¯complex conjugate|such thattterminal point map(eedge¯complex conjugate)=equalswvector}setγfunction(eedge¯complex conjugate) \( T^{*}\mathopen{}\left( γ\right)\mathclose{}\mathopen{}\left( v\right)\mathclose{}= \sum_{\mathopen{}\left\{\, e\,\middle\vert\, i\mathopen{}\left( e\right)\mathclose{}= v\,\right\}\mathclose{}}{}γ\mathopen{}\left( e\right)\mathclose{}-\sum_{\mathopen{}\left\{\, \overline{e}\,\middle\vert\, t\mathopen{}\left( \overline{e}\right)\mathclose{}= w\,\right\}\mathclose{}}{}γ\mathopen{}\left( \overline{e}\right)\mathclose{} \) (see Figure II.A for an example).
What does Tlinear map*(Tlinear map):mapsl2(Vvertex set)→tol2(Vvertex set) \( T^{*}\mathopen{}\left( T\right)\mathclose{} : \mathrm{l}^{0}\mathopen{}\left( V\right)\mathclose{} \to \mathrm{l}^{0}\mathopen{}\left( V\right)\mathclose{} \) do? (Tlinear map*(Tlinear map(ffunction)))(vvertex)=equals∑summation{seteedge|such thatiinitial point map(eedge)=equalsvvertex}set(Tlinear map(ffunction))(eedge)-minus∑summation{seteedge|such thattterminal point map(eedge)=equalsvvertex}set(Tlinear map(ffunction))(eedge)=equals∑summation{seteedge|such thatiinitial point map(eedge)=equalsvvertex}set (ffunction(vvertex)-minusffunction(tterminal point map(eedge))) -minus∑summation{seteedge|such thattterminal point map(eedge)=equalsvvertex}set (ffunction(iinitial point map(eedge))-minusffunction(vvertex)) =equalsffunction(vvertex)times(#cardinality(iinitial point map−1inverse(vvertex))+plus#cardinality(tterminal point map−1inverse(vvertex))⏟degdegree(vvertex))-minus∑summationNeighbors wvector of vvertexffunction(wvector) . \[ \mathopen{}\left( T^{*}\mathopen{}\left( T\mathopen{}\left( f\right)\mathclose{}\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( v\right)\mathclose{}= \sum_{\mathopen{}\left\{\, e\,\middle\vert\, i\mathopen{}\left( e\right)\mathclose{}= v\,\right\}\mathclose{}}{}\mathopen{}\left(T\mathopen{}\left( f\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( e\right)\mathclose{}-\sum_{\mathopen{}\left\{\, e\,\middle\vert\, t\mathopen{}\left( e\right)\mathclose{}= v\,\right\}\mathclose{}}{}\mathopen{}\left(T\mathopen{}\left( f\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( e\right)\mathclose{}= \sum_{\mathopen{}\left\{\, e\,\middle\vert\, i\mathopen{}\left( e\right)\mathclose{}= v\,\right\}\mathclose{}}{} \mathopen{}\left(f\mathopen{}\left( v\right)\mathclose{}-f\mathopen{}\left( t\mathopen{}\left( e\right)\mathclose{}\right)\mathclose{}\right)\mathclose{} -\sum_{\mathopen{}\left\{\, e\,\middle\vert\, t\mathopen{}\left( e\right)\mathclose{}= v\,\right\}\mathclose{}}{} \mathopen{}\left(f\mathopen{}\left( i\mathopen{}\left( e\right)\mathclose{}\right)\mathclose{}-f\mathopen{}\left( v\right)\mathclose{}\right)\mathclose{} = f\mathopen{}\left( v\right)\mathclose{}\mathopen{}\left(\underbrace{{\#}\mathopen{}\left({i}^{-1}\mathopen{}\left( v\right)\mathclose{}\right)\mathclose{}+{\#}\mathopen{}\left({t}^{-1}\mathopen{}\left( v\right)\mathclose{}\right)\mathclose{}}_{\mathop{\text{deg}}\mathopen{}\left( v\right)\mathclose{}}\right)\mathclose{}-\sum_{\text{Neighbors w of v}}{}f\mathopen{}\left( w\right)\mathclose{} \text{.} \] Thus Tlinear map*(Tlinear map(ffunction))=equalsDdiagonal operator(ffunction)-minusAself-adjoint operator(ffunction) \( T^{*}\mathopen{}\left( T\mathopen{}\left( f\right)\mathclose{}\right)\mathclose{}= D\mathopen{}\left( f\right)\mathclose{}-A\mathopen{}\left( f\right)\mathclose{} \) where Ddiagonal operator\( D \) is multiplication by the degree function and Aself-adjoint operator\( A \) is the adjacency operator.
Consider Sshift operator\( S \), the unilateral forward shift on l2(Zintegers+positive elements) \( \mathrm{l}^{0}\mathopen{}\left( {\mathbb{Z}}^{+}\right)\mathclose{} \) defined as Sshift operator(xpositive integer1onexpositive integer2two…)=equals(0zero, xpositive integer1one, xpositive integer2two, …) \( S\mathopen{}\left( {x}_{1}, {x}_{2}, \dotsc\right)\mathclose{}= \mathopen{}\left(0, {x}_{1}, {x}_{2}, \dotsc\right)\mathclose{} \). Note that ‖Sshift operator(xvector)‖=equals‖xvector‖ \( \mathopen{}\left\lVert{}S\mathopen{}\left( \mathbf{x}\right)\mathclose{}\right\rVert\mathclose{}= \mathopen{}\left\lVert{}\mathbf{x}\right\rVert\mathclose{} \), so Sshift operator\( S \) is bounded. 〈Sshift operator*(xvector), yvector〉=equals〈xvector, Sshift operator*(yvector)〉=equals〈(xpositive integer1one, xpositive integer2two, …), (0zero, ypositive integer1one, ypositive integer2two, …)〉=equals∑summationninteger=1one∞infinity ypositive integerninteger¯complex conjugatetimesxpositive integerninteger+plus1one =equals〈(xpositive integer2two, xpositive integer3three, …), (ypositive integer1one, ypositive integer2two, …)〉 . \[ \mathopen{}\left\langle{} S^{*}\mathopen{}\left( \mathbf{x}\right)\mathclose{}, \mathbf{y}\right\rangle\mathclose{}= \mathopen{}\left\langle{}\mathbf{x}, S^{*}\mathopen{}\left( \mathbf{y}\right)\mathclose{}\right\rangle\mathclose{}= \mathopen{}\left\langle{}\mathopen{}\left({x}_{1}, {x}_{2}, \dotsc\right)\mathclose{}, \mathopen{}\left(0, {y}_{1}, {y}_{2}, \dotsc\right)\mathclose{}\right\rangle\mathclose{}= \sum_{n=1}^{\infty}{} \overline{{y}_{n}}{x}_{n+1} = \mathopen{}\left\langle{}\mathopen{}\left({x}_{2}, {x}_{3}, \dotsc\right)\mathclose{}, \mathopen{}\left({y}_{1}, {y}_{2}, \dotsc\right)\mathclose{}\right\rangle\mathclose{} \text{.} \] Thus Sshift operator*(xvector)=equals(xpositive integer2two, xpositive integer3three, …) \( S^{*}\mathopen{}\left( \mathbf{x}\right)\mathclose{}= \mathopen{}\left({x}_{2}, {x}_{3}, \dotsc\right)\mathclose{} \) (backwards shift). Sshift operator*(Sshift operator)=equalsI \( S^{*}\mathopen{}\left( S\right)\mathclose{}= I \), but Sshift operator(Sshift operator*(xvector))=equals(0zero, xpositive integer2two, …)=equalsI-minusPprojection1one≠not equal toI , \[ S\mathopen{}\left( S^{*}\mathopen{}\left( \mathbf{x}\right)\mathclose{}\right)\mathclose{}= \mathopen{}\left(0, {x}_{2}, \dotsc\right)\mathclose{}= I-{P}_{1}\neq I \text{,} \] where Pprojection1one\( {P}_{1} \) is the projection on the first entry ( Pprojection1one(xvector)=equals(xpositive integer1one, 0zero, …) \( {P}_{1}\mathopen{}\left( \mathbf{x}\right)\mathclose{}= \mathopen{}\left({x}_{1}, 0, \dotsc\right)\mathclose{} \)).