The two graph-domain quotients of the local inverse

This editorial note derives the graph-domain consequences of the completed local inverse. Its receiving proofs are Section 14.6, (CS40)–(CS50) and the unchanged example in Section 14.8.5, (CE25)–(CE34). The original lesson construction and the unchanged example remain explicit throughout. These source links require access to the course repository. The complete packaged proofs are the local inverse identities and the original-coefficient example.

All spaces below are complex. The $L^2$ inner product is linear in its first argument. Distribution pairings retain the complex-bilinear convention of U006. The Hilbert adjoint introduced below consequently involves conjugation; it is a different operation from the distributional transpose $p(D)\mapsto p(-D)$ in U006 (CS4).

1. The unchanged polynomial graph space and completed maps

Retain the original nonzero polynomial $p\in\mathbb C[\xi_1,\ldots,\xi_n]$, its degree $m$, the original coordinates, and $D_j=-i\partial_{x_j}$. For every polynomial $q$, keep the full strength array

$$\widetilde q(\xi)^2=\sum_{\alpha\in\mathbb N^n}|\partial_\xi^\alpha q(\xi)|^2, \qquad V_p=\{q:\widetilde q(\xi)\leq C_q\widetilde p(\xi) \text{ for every }\xi\in\mathbb R^n\text{ and some finite }C_q\}, \qquad \|q\|_p=\sup_{\xi\in\mathbb R^n}\frac{\widetilde q(\xi)}{\widetilde p(\xi)}. \tag{GM1}$$

Every zeroth and higher derivative, including its zero value, remains part of this definition. To recall why its denominator is positive, choose a nonzero degree-$m$ coefficient $a_{\alpha_0}$. Then $\partial^{\alpha_0}p=\alpha_0!a_{\alpha_0}$, so $\widetilde p\geq\alpha_0!|a_{\alpha_0}|>0$. For $m=0$, this bound is $|p|>0$. It follows that $1,p\in V_p$, with $\|p\|_p=1$. The pointwise triangle inequality for the complete derivative arrays makes $V_p$ a vector space and $\|\cdot\|_p$ a norm.

This vector space is finite dimensional. Indeed $\widetilde p(\xi)\leq A(1+|\xi|)^m$ for some finite $A$, since its derivative array is finite and each entry is a polynomial of degree at most $m$. If $q\in V_p$ had degree $d>m$, its degree-$d$ part would be nonzero at some real vector $v$. A polynomial zero on every real vector has every coefficient zero: apply the one-variable root bound in the first variable for each fixed choice of the other variables and repeat by induction. Consequently $q(tv)=t^d q_d(v)+O(t^{d-1})$ would contradict $|q(tv)|\leq\|q\|_p A(1+t|v|)^m$. Thus $V_p$ lies in the finite-dimensional space of polynomials of degree at most $m$. These are also the arguments in U006 (CS5)–(CS6).

Choose the actual basis $q_1,\ldots,q_s$ used for the graph norm. Retain the original polynomials $p_\nu\in V_p$, coefficients $c_\nu\in C(\Omega)$ with $c_\nu(x_0)=0$, and the bounded open neighborhood $U$ of (CS42). In particular the coefficients are bounded on $U$, because $\overline U\subset\Omega$ is compact. Put $Y=L^2(U)$, and use exactly the full common maximal graph space

$$H=\mathcal H_p(U)=\{u\in L^2(U):q_j(D)u\in L^2(U),\ 1\leq j\leq s\}, \qquad \|u\|_H^2=\|u\|_2^2+\sum_{j=1}^s\|q_j(D)u\|_2^2. \tag{GM2}$$

The derivatives in this definition are distributional. Its inner product is the corresponding sum of the original $L^2$ inner products. Completeness follows directly: if $u_k$ is Cauchy in this norm, let $u$ and $v_j$ be the $L^2$ limits of $u_k$ and $q_j(D)u_k$. Testing against a compactly supported smooth function shows $q_j(D)u=v_j$, because differentiation is continuous in the distributional topology and $L^2$ convergence implies distributional convergence by Cauchy–Schwarz. Thus $u\in H$ and $u_k\to u$ in $H$. A basis expansion proves that the set $H$ contains the $L^2$ distributional derivative for every $q\in V_p$. No graph norm is changed when this fact is used.

Write the exact basis expansions

$$p=\sum_{j=1}^s\alpha_jq_j, \qquad p_\nu=\sum_{j=1}^s\beta_{\nu j}q_j, \qquad g_j(x)=\alpha_j+\sum_{\nu=1}^r c_\nu(x)\beta_{\nu j}. \tag{GM3}$$

The graph-domain realization of the original operator is

$$P:H\longrightarrow Y, \qquad Pu=p(D)u+\sum_{\nu=1}^r c_\nu p_\nu(D)u =\sum_{j=1}^s g_jq_j(D)u. \tag{GM4}$$

Every coefficient multiplies an already defined $L^2$ function. Define the finite bound

$$C_P=\mathop{\rm ess\,sup}_{x\in U} \left(\sum_{j=1}^s|g_j(x)|^2\right)^{1/2}. \tag{GM5}$$

Pointwise Cauchy–Schwarz followed by integration proves $\|Pu\|_2\leq C_P\|u\|_H$. This is an explicit bound, with every original coefficient present; sharpness of the operator norm is not claimed.

Keep the completed maps $S_U,B_U,T_U^{-1}$, and $E=S_UT_U^{-1}$ from (CS38)–(CS50), rather than constructing a new inverse. Keep

$$K_0=K_{m,n,R_0},\qquad b=K_0\sum_{\nu=1}^r\|p_\nu\|_p\sup_U|c_\nu|<1, \qquad C_E=\frac{K_0}{1-b} \left(\|1\|_p^2+\sum_{j=1}^s\|q_j\|_p^2\right)^{1/2}. \tag{GM6}$$

Equations (CS41), (CS45), and (CS47) give the bounded map $E:Y\to H$ with $\|Ef\|_H\leq C_E\|f\|_2$. Equation (CS48) proves $PE=I_Y$; equation (CS50) proves $EPu=u$ for $u\in C_c^\infty(U)$. These are completed input results, with their full proofs retained in the input lesson. This note derives their domain consequences without adding a solvability premise or suppressing any part of their construction.

2. The minimal domain, its closed image, and the exact kernel projection

Define the actual minimal common graph domain by

$$H_{\min}=\overline{C_c^\infty(U)}^{\|\cdot\|_H}, \qquad M=P(H_{\min})\subset Y, \qquad N=\ker(P:H\to Y). \tag{GM7}$$

Theorem GM-A. The identity $EP=I$ holds on $H_{\min}$. The restriction $P_{\min}:H_{\min}\to M$ is a bounded bijection with inverse $E|_M$, and

$$\|u\|_H\leq C_E\|Pu\|_2\quad(u\in H_{\min}), \qquad E(M)=H_{\min}, \qquad N\cap H_{\min}=\{0\}. \tag{GM8}$$

The image $M$ is closed in $Y$. The exact operator

$$Q=I_H-EP:H\to H \tag{GM9}$$

is a bounded projection onto $N$, with kernel $E(Y)$, and

$$Q^2=Q,\quad PQ=0,\quad QE=0,\quad Q|_N=I_N,\quad \|Q\|_{H\to H}\leq 1+C_EC_P. \tag{GM10}$$

It vanishes on $H_{\min}$. The completed solution space is the exact topological direct sum

$$H=N\mathbin{\dotplus}E(Y),\qquad u=Qu+E(Pu). \tag{GM11}$$

For every $f\in Y$, all solutions in this original graph domain, and only those solutions, are

$$\{u\in H:Pu=f\}=Ef+N. \tag{GM12}$$

Proof. For $u\in H_{\min}$, choose $u_k\in C_c^\infty(U)$ with $u_k\to u$ in $H$. Boundedness of $P$ gives $Pu_k\to Pu$ in $Y$. Boundedness of $E:Y\to H$ then gives $EPu_k\to EPu$ in $H$. On the compactly supported functions the input identity $EPu_k=u_k$ is initially an equality in $L^2$, and hence an equality in $H$, since the equal $L^2$ classes have the same distributional derivatives. Taking the limit proves $EPu=u$ in $H$. The estimate in (GM8) follows. It proves injectivity of $P_{\min}$; surjectivity onto $M$ is its definition. If $f=Pu\in M$ with $u\in H_{\min}$, then $Ef=u\in H_{\min}$. Conversely $u=E(Pu)$ for every $u\in H_{\min}$. Thus $E(M)=H_{\min}$, and $N\cap H_{\min}=0$.

To prove closedness, let $f_k\in M$ converge in $Y$ to $f$. Write $u_k=Ef_k\in H_{\min}$. The bound for $E$ gives $u_k\to Ef$ in $H$. Since $H_{\min}$ is closed, $Ef\in H_{\min}$; since $PEf=f$, it follows that $f\in M$. This proves the claimed closedness without a Fredholm or ellipticity assumption.

The identity $PE=I_Y$ gives $(EP)^2=EP$. Expanding $(I-EP)^2$ proves $Q^2=Q$. It also gives $PQ=P-PEP=0$ and $QE=E-EPE=0$. If $v\in N$, then $Qv=v$, so the range of $Q$ is exactly $N$. If $Qu=0$, then $u=E(Pu)\in E(Y)$; if $u=Ef$, then $Qu=0$. Thus the kernel is exactly $E(Y)$. Both summands are closed: $N$ is the kernel of a bounded map, and $E(Y)$ is the kernel of $Q$. The intersection is zero, because $PEf=0$ implies $f=0$. Formula (GM11) is the definition of $Q$. Its projections are $Q$ and $EP$, so the direct sum is topological. The norm bound in (GM10) is the triangle inequality and the two operator bounds. Finally $P(Ef+v)=f$ for $v\in N$; any solution $u$ satisfies $P(u-Ef)=0$. This proves (GM12). ∎

Corollary GM-A1: the single-operator minimal closure. Consider $P$ on $C_c^\infty(U)\subset L^2(U)$ with its original expression (GM4). Its closure as an unbounded operator in $L^2(U)$ has domain exactly $H_{\min}$, and it is injective with closed range $M$. On this domain the original full graph norm and the single-operator graph norm obey

$$\|u\|_H\leq C_E\left(\|u\|_2^2+\|Pu\|_2^2\right)^{1/2}, \qquad \left(\|u\|_2^2+\|Pu\|_2^2\right)^{1/2} \leq (1+C_P^2)^{1/2}\|u\|_H. \tag{GM13}$$

Proof. The second estimate holds on all of $H$ by (GM2) and (GM5); the first follows from (GM8). If $u_k\in C_c^\infty(U)$ converges in the single-operator graph norm, the difference version of (GM8) makes it Cauchy in the full original $H$ norm. Its $H$ limit is in $H_{\min}$ and agrees with its $L^2$ limit, and boundedness of $P$ identifies the limiting image. Conversely the defining $H$-approximation for $H_{\min}$ gives approximation in the single-operator graph norm by the second estimate. This proves equality of the two closures while retaining both norms and their exact comparison.

For closedness as an unbounded operator, suppose $u_k\in H_{\min}$, $u_k\to u$ in $L^2$, and $Pu_k\to f$ in $L^2$. Again (GM8) makes $u_k$ Cauchy in $H$. Its limit belongs to $H_{\min}$, agrees with $u$, and has image $f$. The domain is dense in $L^2(U)$, because it contains $C_c^\infty(U)$, which is dense: truncate an $L^2$ function to compact subsets a positive distance from the boundary, then mollify with radii smaller than that distance. The truncations converge by dominated convergence, and the mollifications by translation continuity in $L^2$. These operations are also proved in the prerequisite cited in U006 before (CS38). Injectivity and closed range were proved in GM-A. ∎

The word “maximal” in (GM2) refers to all original constant-polynomial derivatives in $V_p$. We have not replaced that domain by a domain that asks only for $Pu\in L^2$. For continuous coefficients, multiplying an arbitrary distributional derivative by a coefficient is not automatically defined. The exact closure statement just proved applies to the minimal operator and supplies the bridge between its two graph norms without altering the maximal common domain.

3. The complete quotient maps, constants, and three projections

Equip $H/H_{\min}$ and $Y/M$ with their actual quotient norms:

$$\|[u]\|_{H/H_{\min}}=\inf_{h\in H_{\min}}\|u-h\|_H, \qquad \|[f]\|_{Y/M}=\inf_{m\in M}\|f-m\|_2. \tag{GM14}$$

On $N\oplus(Y/M)$, use the Hilbert direct-sum norm

$$\|(v,[f])\|_\oplus^2=\|v\|_H^2+\|[f]\|_{Y/M}^2. \tag{GM15}$$

Theorem GM-B. The following explicit maps are inverse bounded linear isomorphisms:

$$\Phi:H/H_{\min}\longrightarrow N\oplus(Y/M), \qquad \Phi([u])=(Qu,[Pu]), \tag{GM16}$$

$$\Psi:N\oplus(Y/M)\longrightarrow H/H_{\min}, \qquad \Psi(v,[f])=[v+Ef]. \tag{GM17}$$

Put $C_Q=1+C_EC_P$. Their original-norm bounds are

$$\|\Phi\|\leq(C_Q^2+C_P^2)^{1/2}, \qquad \|\Psi\|\leq(1+C_E^2)^{1/2}. \tag{GM18}$$

The exact quotient norm, before using either estimate, is

$$\|\Psi(v,[f])\|_{H/H_{\min}} =\inf_{m\in M}\|v+E(f-m)\|_H. \tag{GM19}$$

There is also the split exact sequence with all original connecting maps

$$0\longrightarrow N \xrightarrow{v\mapsto[v]}H/H_{\min} \xrightarrow{[u]\mapsto[Pu]}Y/M \longrightarrow0, \qquad [f]\mapsto[Ef] \text{ is a bounded splitting.} \tag{GM20}$$

Proof. If $u$ is replaced by $u+h$ for $h\in H_{\min}$, then $Qh=0$ and $Ph\in M$, so (GM16) is well defined. If $f$ is replaced by $f+m$ for $m\in M$, then $Em\in H_{\min}$, so (GM17) is well defined. Using $QE=0$, $Qv=v$, $Pv=0$, and $PEf=f$, we obtain $\Phi\Psi(v,[f])=(v,[f])$. Using (GM11), we obtain $\Psi\Phi([u])=[Qu+EPu]=[u]$.

For every $h\in H_{\min}$, $$\|Qu\|_H\leq C_Q\|u-h\|_H, \qquad \|[Pu]\|_{Y/M}\leq\|P(u-h)\|_2\leq C_P\|u-h\|_H.$$ Square, add, and take the infimum to obtain the first bound. For every $m\in M$, $$\|[v+Ef]\|_{H/H_{\min}} \leq\|v+E(f-m)\|_H \leq\|v\|_H+C_E\|f-m\|_2.$$ Taking the infimum and then applying two-variable Cauchy–Schwarz proves the second bound. Since $H_{\min}=E(M)$, the infimum in the definition of the quotient norm is exactly (GM19), not a newly imposed norm. The injection in (GM20) is injective because $N\cap H_{\min}=0$. The second map is surjective because $PE=I$. If $[Pu]=0$, then $Pu\in M$, so $EPu\in H_{\min}$ and $[u]=[Qu]$ is in the displayed injection. The splitting follows from $PE=I$ and is bounded by $C_E$. Thus every assertion of exactness and splitting is established. ∎

For completeness, a closed subspace $M$ of a Hilbert space has an orthogonal projection with norm at most one. Here is the argument used below. For $f\in Y$, let $d=\inf_{m\in M}\|f-m\|_2$, and choose $m_k\in M$ approaching that infimum. The parallelogram identity gives $$\|m_k-m_\ell\|_2^2 \leq 2\|f-m_k\|_2^2+2\|f-m_\ell\|_2^2-4d^2\longrightarrow0.$$ Thus $m_k\to m\in M$. The real one-variable variation of $\|f-m-tz\|^2$, first for $z\in M$, then for $iz$, proves $f-m\perp M$. This orthogonal decomposition is unique, depends linearly on $f$, and its two squared component norms add to $\|f\|_2^2$; consequently each projection has norm at most one. Write these projections as $\Pi_M$ and $\Pi_{M^\perp}$.

There are three actual bounded projections on $H$:

$$Q=I-EP, \qquad R_{\min}=E\Pi_M P, \qquad R_{\perp}=E\Pi_{M^\perp}P. \tag{GM21}$$

Their ranges are respectively $N,H_{\min},E(M^\perp)$, their pairwise products in both orders are zero, and their sum is $I_H$. Indeed $PE=I$, the two orthogonal $Y$-projections square to themselves and have zero products, and $PQ=QE=0$. The range of $R_{\min}$ is $E(M)=H_{\min}$, and it is the identity there because $EP=I$ on that domain. The same calculation proves the other range and identity. Thus

$$H=H_{\min}\mathbin{\dotplus}N\mathbin{\dotplus}E(M^\perp), \qquad \|R_{\min}\|,\|R_{\perp}\|\leq C_EC_P. \tag{GM22}$$

These sums use the original $H$ norm. Orthogonality between their $H$-summands is not asserted. The precise quotient-coordinate map is also $[u]\mapsto(Qu,\Pi_{M^\perp}Pu)$, because $[f]\mapsto\Pi_{M^\perp}f$ is an isometry from $Y/M$ onto $M^\perp$.

4. The second factor is the exact adjoint obstruction space

Define the original Hilbert formal adjoint distribution on $v\in L^2(U)$ by

$$P^\dagger v =\overline p(D)v +\sum_{\nu=1}^r\overline{p_\nu}(D) (\overline{c_\nu}\,v), \qquad N^\dagger=\{v\in L^2(U):P^\dagger v=0\text{ in }\mathcal D'(U)\}. \tag{GM23}$$

The bars on polynomials conjugate their coefficients, not their variables. Each product inside a derivative is an $L^2$ function; distributional differentiation of that product is defined even when $c_\nu$ is only continuous. No derivative of a continuous coefficient is postulated as a function.

Theorem GM-C. One has $M^\perp=N^\dagger$. In particular the quotient factor $Y/M$ is isometrically isomorphic to the exact distributional adjoint kernel, and the original full graph-domain quotient has the bounded isomorphism

$$H/H_{\min}\longrightarrow N\oplus N^\dagger, \qquad [u]\longmapsto(Qu,\Pi_{N^\dagger}Pu), \qquad (v,w)\longmapsto[v+Ew] \tag{GM24}$$

with the bounds in (GM18), using the $H$ norm on $N$ and the $L^2$ norm on $N^\dagger$.

Proof. For $u\in C_c^\infty(U)$, integration by parts or the definition of distributional derivatives gives

$$\langle Pu,v\rangle_{L^2} =\overline{\langle P^\dagger v,\overline u\rangle_{\mathcal D',\mathcal D}}. \tag{GM25}$$

For the signs, $D_j=-i\partial_j$, its distributional transpose is $-D_j$, and $-D_j\overline u=\overline{D_ju}$. The same identity term by term in the complete polynomials gives $\overline p(D)$ on $v$ and $\overline{p_\nu}(D)$ on $\overline{c_\nu}v$. There is no boundary contribution because $u$ is compactly supported.

If $v\perp M$, its pairing with $Pu$ vanishes for every such $u$, so (GM25) proves $P^\dagger v=0$. Conversely, if $P^\dagger v=0$, the pairing vanishes on $C_c^\infty(U)$. For any $u\in H_{\min}$, choose the original graph approximation $u_k\to u$. Then $Pu_k\to Pu$ in $L^2$, so Cauchy–Schwarz extends the zero pairing to $Pu$. Hence $v\perp M$. This proves equality. Since $M$ is closed, the orthogonal quotient identification proved after GM-B applies. Substitution in (GM16)–(GM17) gives (GM24) and its inverse, and all bounds are unchanged. ∎

The unbounded adjoint of the closed densely defined minimal operator of GM-A1 has the complete domain

$$\operatorname{Dom}(P_{\min}^*) =\{v\in L^2(U):P^\dagger v\in L^2(U)\text{ as a distribution}\}, \qquad P_{\min}^*v=P^\dagger v. \tag{GM26}$$

To prove this, the defining adjoint equality on $H_{\min}$ restricts to (GM25), and so identifies its representing $L^2$ function with $P^\dagger v$. Conversely an $L^2$ representing function for $P^\dagger v$ gives the adjoint equality first on the compactly supported functions by (GM25), then on $H_{\min}$ by graph approximation; both $u_k\to u$ and $Pu_k\to Pu$ in $L^2$. This proves both inclusions of the domain and the displayed operator formula. Therefore the space $N^\dagger$ in (GM24) is exactly $\ker P_{\min}^*$, with no additional regularity condition.

There is an exact existence criterion for the minimal-domain solution: for $f\in Y$, a solution $u\in H_{\min}$ of $Pu=f$ exists if and only if $\langle f,v\rangle_2=0$ for every $v\in N^\dagger$. In that event its unique value is $u=Ef$. Indeed a minimal-domain image lies in $M$ and is orthogonal to $M^\perp=N^\dagger$. Conversely the orthogonal decomposition $f=\Pi_Mf+\Pi_{M^\perp}f$, proved above, shows that orthogonality to $M^\perp$ forces the second component to vanish, so $f\in M$. Equation (GM8) then puts $Ef$ in $H_{\min}$, and $PEf=f$. Injectivity of $P_{\min}$ proves uniqueness. Thus the second quotient factor specifies the exact right-hand-side obstruction and its receiving map, while the first specifies the ambiguity among full graph-domain solutions.

5. The unchanged example and its entire kernel modes

Now retain every quantity of U006 (CE1)–(CE34):

$$p(\xi_1,\xi_2)=\xi_1^2+i\xi_2, \qquad n=2,\quad m=2,\quad x_0=(0,0), \qquad U=(-1/4,1/4)^2, \tag{GM27}$$

$$\widetilde p(\xi)^2 =|\xi_1^2+i\xi_2|^2+|2\xi_1|^2+|i|^2+|2|^2 =\xi_1^4+\xi_2^2+4\xi_1^2+1+4, \tag{GM28}$$

$$C=\begin{pmatrix}14&-20&7\\-20&32&-12\\7&-12&5\end{pmatrix}, \quad\lambda_*=\text{the largest eigenvalue of }C, \quad K=41472\lambda_*\sqrt{33}\,e^2, \quad a=\sqrt{\frac{\sqrt2-1}{2}}, \quad\varepsilon=\frac2{Ka}>0, \quad c(x)=\varepsilon x_1. \tag{GM29}$$

Here $\lambda_*$ is the matrix eigenvalue from U006 (CE7)–(CE10). It is not the freely varying mode parameter $\lambda$ below. The full original differential expression becomes

$$P=D_1^2+iD_2+\varepsilon x_1D_1 =-\partial_1^2+\partial_2-i\varepsilon x_1\partial_1. \tag{GM30}$$

Indeed $(-i)^2=-1$, $i(-i)=1$, and $\varepsilon x_1(-i)=-i\varepsilon x_1$. Thus every sign and the nonzero drift coefficient are retained.

Lemma GM-D1: the full weaker-polynomial space. For this actual $p$,

$$V_p=\operatorname{span}_{\mathbb C}\{1,\xi_1,\xi_1^2,\xi_2\}, \quad \|1\|_p=1/\sqrt5, \quad\|\xi_1\|_p=a, \quad\|\xi_1^2\|_p=1, \quad\|\xi_2\|_p=1. \tag{GM31}$$

Proof. The degree argument after (GM1) puts every $q\in V_p$ in degree at most two. Write its complete expression $$q=A_{20}\xi_1^2+A_{11}\xi_1\xi_2+A_{02}\xi_2^2 +A_{10}\xi_1+A_{01}\xi_2+A_{00}.$$ At $(0,t)$, the bound $|q(0,t)|\leq C_q\sqrt{t^2+1+4}$ forces $A_{02}=0$ by dividing by $t^2$ and taking $t\to+\infty$. At $(t,t^2)$, the complete denominator is $\sqrt{t^4+t^4+4t^2+1+4}$; dividing the bound by $t^3$ then forces $A_{11}=0$. All remaining coefficients are unrestricted once the four displayed monomials are in $V_p$.

For $1$, the complete strength numerator is $1$, whose ratio is maximal at $(0,0)$, where the denominator is $\sqrt{1+4}=\sqrt5$. For $\xi_1$, the squared ratio is $$\frac{\xi_1^2+1}{\xi_1^4+\xi_2^2+4\xi_1^2+1+4}.$$ It is maximal at $\xi_2=0$. Put $t=\xi_1^2\geq0$. Differentiating $(t+1)/(t^2+4t+1+4)$ gives numerator $-t^2-2t+1$; it is positive before $t=\sqrt2-1$ and negative afterwards. Substitution gives $(\sqrt2-1)/2=a^2$, including the actual endpoint value $1/5$ and the limit $0$. For $\xi_1^2$, the squared ratio is $$\frac{\xi_1^4+4\xi_1^2+4}{\xi_1^4+\xi_2^2+4\xi_1^2+1+4}\leq1,$$ and its values at $(t,0)$ tend to one. For $\xi_2$, the squared ratio is $$\frac{\xi_2^2+1}{\xi_1^4+\xi_2^2+4\xi_1^2+1+4}\leq1,$$ and its values at $(0,t)$ tend to one. Thus the two suprema are exactly one. The derivative-array triangle inequality shows that every linear combination of these four monomials is in $V_p$; their polynomial independence makes them a basis. ∎

For the explicit norm calculation choose the basis $(1,\xi_1,\xi_1^2,\xi_2)$. This is a declared instance of the basis in (GM2), not a replacement of $p$. The graph set and its full norm are

$$H=\{u\in L^2(U):D_1u,D_1^2u,D_2u\in L^2(U)\}, \quad \|u\|_H^2=\|u\|_2^2+\|u\|_2^2 +\|D_1u\|_2^2+\|D_1^2u\|_2^2+\|D_2u\|_2^2. \tag{GM32}$$

In this exact basis the full coefficient vector of $P$ is $(0,\varepsilon x_1,1,i)$. Hence

$$C_P=\left(|0|^2+\varepsilon^2/16+|1|^2+|i|^2\right)^{1/2}, \qquad b=Ka\sup_U|c|=Ka\frac{\varepsilon}{4}=\frac12, \quad C_E=2K\left(\frac15+\frac15+a^2+1+1\right)^{1/2}. \tag{GM33}$$

The supremum for $C_P$ is the essential supremum over the unchanged open square; values approach its endpoints on sets of positive measure. The two $1/5$ terms in $C_E$ are respectively the separate $\|1\|_p^2$ term of the graph estimate and the $q_1=1$ contribution. The two $1$ terms belong to $\xi_1^2$ and $\xi_2$. None is absorbed or omitted. Thus all results GM-A–GM-C apply with the constants (GM29) and (GM33).

Theorem GM-D2: the requested even modes. For every $\lambda\in\mathbb C$, put the empty product equal to one and define

$$c_k(\lambda)=\frac{\prod_{h=0}^{k-1}(\lambda-2i\varepsilon h)}{(2k)!}, \qquad \varphi_\lambda(z)=\sum_{k=0}^\infty c_k(\lambda)z^{2k}, \qquad u_\lambda(x_1,x_2)=e^{\lambda x_2}\varphi_\lambda(x_1). \tag{GM34}$$

The series and every derivative converge uniformly on every compact subset of $\mathbb C^2$. The functions are entire, lie in the full actual space $H$, and satisfy $Pu_\lambda=0$. Any finite family with distinct parameters is linearly independent. In particular $N$ and $H/H_{\min}$ are infinite dimensional, and every $u_\lambda$ represents a nonzero class in $H/H_{\min}$.

Proof of convergence and derivatives. Put $A_\lambda=|\lambda|+2\varepsilon>0$. For $h\geq0$, $$|\lambda-2i\varepsilon h| \leq|\lambda|+2\varepsilon h \leq A_\lambda(h+1).$$ Therefore the full product satisfies $$|c_k(\lambda)|\leq\frac{A_\lambda^k k!}{(2k)!} \leq\frac{A_\lambda^k}{k!}, \qquad |\varphi_\lambda(z)|\leq e^{A_\lambda|z|^2}. \tag{GM35}$$ For the second inequality, $(2k)!\geq(k!)^2$, since $\prod_{j=1}^k(k+j)\geq\prod_{j=1}^k j$. If $r\geq0$, the $r$-th derivative of the $k$-th term is zero for $2k<r$, and otherwise has magnitude at most $$(2k)^r\frac{A_\lambda^k}{k!}R^{2k} \quad\text{on }|z|\leq\rho,\quad R=\max(1,\rho).$$ For $r=0$, use $0^0=1$ in this majorant. The majorant series converges: for $k\geq1$ the ratio of successive terms is $$A_\lambda R^2\frac{(1+1/k)^r}{k+1}\longrightarrow0.$$ It proves uniform convergence of each differentiated series on the compact disk. Termwise differentiation follows, for example, by integrating the uniformly convergent derivative series along a line segment and using the value at a fixed point; iteration handles each derivative. The factor $e^{\lambda z_2}$ is entire with $s$-th derivative $\lambda^s e^{\lambda z_2}$. On $|z_2|\leq\rho_2$ this has magnitude at most $|\lambda|^s e^{|\lambda|\rho_2}$, using $|\lambda|^0=1$ also at $\lambda=0$. Multiplying the two compact majorants proves uniform convergence and differentiation of every mixed derivative on every compact polydisk, and therefore the asserted entire extension.

Proof of the equation. The exact coefficients obey $$(2k+2)(2k+1)c_{k+1}(\lambda) =(\lambda-2i\varepsilon k)c_k(\lambda). \tag{GM36}$$ If a coefficient is zero this equality still holds, since it was derived by multiplication of the complete product and factorial, not by division by that coefficient. By the proved termwise differentiation, $$\varphi_\lambda''(z) =\sum_{k\geq0}(\lambda-2i\varepsilon k)c_k z^{2k} =\lambda\varphi_\lambda(z)-i\varepsilon z\varphi_\lambda'(z). \tag{GM37}$$ Substituting the full derivatives in (GM30) gives $$Pu_\lambda=e^{\lambda x_2} \big[-\varphi_\lambda''+\lambda\varphi_\lambda -i\varepsilon x_1\varphi_\lambda'\big]=0. \tag{GM38}$$

Proof of membership and explicit bounds. On the closed actual square $|x_1|,|x_2|\leq1/4$, let $$F_\lambda=\exp\left(|\lambda|/4+A_\lambda/16\right).$$ The positive coefficient majorant $\sum A_\lambda^k|x_1|^{2k}/k!=e^{A_\lambda|x_1|^2}$ gives, including its differentiated sums, $$|u_\lambda|\leq F_\lambda, \quad |D_1u_\lambda|\leq(A_\lambda/2)F_\lambda, \quad |D_1^2u_\lambda|\leq(2A_\lambda+A_\lambda^2/4)F_\lambda, \quad |D_2u_\lambda|\leq|\lambda|F_\lambda. \tag{GM39}$$ For the first derivative, differentiate the positive majorant to obtain $2A_\lambda\rho e^{A_\lambda\rho^2}$; for the second, obtain $(2A_\lambda+4A_\lambda^2\rho^2)e^{A_\lambda\rho^2}$, and use the actual $\rho=1/4$. The complex unit factors in $D_j$ do not change these absolute values. Since $|U|=(1/2)(1/2)=1/4$, the original norm obeys $$\|u_\lambda\|_H^2\leq\frac14 F_\lambda^2 \left[1+1+(A_\lambda/2)^2 +(2A_\lambda+A_\lambda^2/4)^2+|\lambda|^2\right]<\infty. \tag{GM40}$$ Thus the classical derivatives represent the required distributional derivatives and $u_\lambda\in H$. This also proves membership for any other declared basis, because every basis polynomial belongs to the four-dimensional space proved in GM-D1 and has a finite expansion. The norm for that basis stays its own original sum.

Proof of independence. Suppose $\sum_{j=1}^J b_j u_{\lambda_j}=0$ in $L^2(U)$, with distinct $\lambda_j$. The sum is continuous, so it vanishes at every point of $U$: a nonzero value would have a neighborhood of positive measure with nonzero values. At $x_1=0$, $\varphi_{\lambda_j}(0)=1$, giving $\sum_j b_j e^{\lambda_jx_2}=0$ for $|x_2|<1/4$. Differentiate in $x_2$ at zero for orders $\ell=0,\ldots,J-1$. Then $\sum_j b_j\lambda_j^\ell=0$. The Vandermonde determinant is $\prod_{i<j}(\lambda_j-\lambda_i)\neq0$; its determinant formula follows by viewing the determinant as an alternating polynomial, dividing by every displayed difference, and comparing the coefficient of $\lambda_2\lambda_3^2\cdots\lambda_J^{J-1}$. Hence all $b_j=0$. There are arbitrarily large such families, so $N$ is infinite dimensional. Every mode is nonzero because $u_\lambda(0,0)=1$. Its intersection with $H_{\min}$ is zero by (GM8). The injection in (GM20) preserves every finite independence relation in the quotient. ∎

6. Explicit adjoint modes and the size of both quotient factors

For the unchanged smooth example (GM30), the full Hilbert formal adjoint is

$$P^\dagger=-\partial_1^2-\partial_2 -i\varepsilon x_1\partial_1-i\varepsilon. \tag{GM41}$$

The last zeroth-order term is indispensable. The adjoint of $-i\varepsilon x_1\partial_1$ is $-i\varepsilon\partial_1(x_1\,\cdot)$, by integration by parts with the conjugated coefficient; the product rule produces exactly the last two terms in (GM41).

Theorem GM-E. The functions

$$v_\lambda(x_1,x_2) =e^{-(\lambda+i\varepsilon)x_2}\varphi_\lambda(x_1), \qquad \lambda\in\mathbb C, \tag{GM42}$$

are entire, belong to $H$ and to $N^\dagger$, and every finite family with distinct parameters is linearly independent. Consequently $M\neq Y$, $Y/M\cong N^\dagger$ is infinite dimensional, and the two separate factors in (GM24) are both infinite dimensional. In particular the closed minimal operator is injective with infinite-dimensional cokernel; the completed maximal common graph operator is surjective with infinite-dimensional kernel. Neither is Fredholm.

Proof. The compact convergence proof for $\varphi_\lambda$ is unchanged. The exponential factor is entire and its $s$-th derivative is $(-(\lambda+i\varepsilon))^s$ times itself. Replace $F_\lambda$ in (GM39)–(GM40) by $$F^\dagger_\lambda =\exp\left(|\lambda+i\varepsilon|/4+A_\lambda/16\right)$$ and replace the last derivative factor $|\lambda|$ by $|\lambda+i\varepsilon|$. The original $H$ norm still has both separate $L^2$ contributions and all three derivative contributions, so this proves membership without dropping a term. Put $\mu=-(\lambda+i\varepsilon)$. Substitution of (GM37) into the complete adjoint expression gives $$P^\dagger v_\lambda=e^{\mu x_2} \big[-\varphi_\lambda''-\mu\varphi_\lambda -i\varepsilon x_1\varphi_\lambda'-i\varepsilon\varphi_\lambda\big] =e^{\mu x_2}(-\lambda-\mu-i\varepsilon)\varphi_\lambda=0. \tag{GM43}$$ This uses the unchanged drift and the exact zeroth-order adjoint term. At $x_1=0$ the functions restrict to $e^{-(\lambda+i\varepsilon)x_2}$. Distinct $\lambda$ give distinct exponents, and the preceding continuity and Vandermonde proof proves their finite independence. Each $v_\lambda$ is therefore a nonzero element of $N^\dagger=M^\perp$, proving both the properness and infinite codimension of $M$. The other statements follow from GM-A and GM-D2. A bounded operator between Hilbert spaces is Fredholm only when it has closed range and finite-dimensional kernel and cokernel; the unbounded minimal operator has the same criterion in its graph domain. The infinite dimensions just proved rule this out in each case. ∎

There are explicit polynomial members in both entire families. If $\lambda=2i\varepsilon j$, $j\in\mathbb N$, then the factor with $h=j$ vanishes and

$$\varphi_{2i\varepsilon j}(x_1) =\sum_{k=0}^j\frac{\prod_{h=0}^{k-1}(2i\varepsilon j-2i\varepsilon h)}{(2k)!}x_1^{2k}, \qquad [x_1^{2j}]\varphi_{2i\varepsilon j} =\frac{(2i\varepsilon)^j j!}{(2j)!}\neq0. \tag{GM44}$$

The factors before that first zero are nonzero; hence the degree is exactly $2j$. Thus $$u_0=1,\quad v_0=e^{-i\varepsilon x_2},\qquad u_{2i\varepsilon}=e^{2i\varepsilon x_2}(1+i\varepsilon x_1^2), \quad v_{2i\varepsilon}=e^{-3i\varepsilon x_2}(1+i\varepsilon x_1^2). \tag{GM45}$$ These formulas exhibit the constant mode and a nonconstant mode with the original nonzero $\varepsilon$. The constant mode is outside $H_{\min}$. It follows that compactly supported smooth functions are not dense in the full maximal graph norm. More quantitatively, for every $v\in N$ and $h\in H_{\min}$, $Q(v-h)=v$, so

$$\frac{\|v\|_H}{1+C_EC_P} \leq\|[v]\|_{H/H_{\min}}\leq\|v\|_H. \tag{GM46}$$

In particular (GM32) gives $\|u_0\|_H^2=|U|+|U|=1/4+1/4$, so the distance of $1$ from $H_{\min}$ is at least $\sqrt{1/4+1/4}/(1+C_EC_P)>0$. This is an explicit obstruction in the unchanged graph metric, rather than an unspecified boundary interpretation.

7. The second initial value and odd modes

The even series fixes $\varphi_\lambda(0)=1$, $\varphi_\lambda'(0)=0$. The complementary initial value has the exact series

$$\psi_\lambda(z) =\sum_{k=0}^\infty \frac{\prod_{h=0}^{k-1}(\lambda-i\varepsilon(2h+1))}{(2k+1)!}z^{2k+1}. \tag{GM47}$$

With $A^o_\lambda=|\lambda|+3\varepsilon$, every product factor is at most $A^o_\lambda(h+1)$ in modulus. Consequently the coefficient is bounded by $(A^o_\lambda)^k/k!$, since $(2k+1)!\geq(k!)^2$. Differentiating a term $r$ times on a compact disk gives a majorant $(2k+1)^r(A^o_\lambda)^k R^{2k+1}/k!$ with $R=\max(1,\rho)$, whose successive ratio tends to zero. The complete derivative and entire-function proof of GM-D2 therefore applies. Its exact recurrence is

$$(2k+3)(2k+2)d_{k+1} =(\lambda-i\varepsilon(2k+1))d_k, \quad d_k=\frac{\prod_{h=0}^{k-1}(\lambda-i\varepsilon(2h+1))}{(2k+1)!}. \tag{GM48}$$

Hence $\psi_\lambda''=\lambda\psi_\lambda-i\varepsilon z\psi_\lambda'$, with $\psi_\lambda(0)=0$ and $\psi_\lambda'(0)=1$. The functions $e^{\lambda x_2}\psi_\lambda(x_1)$ lie in $N$, and $e^{-(\lambda+i\varepsilon)x_2}\psi_\lambda(x_1)$ lie in $N^\dagger$, by the same full substitutions (GM38) and (GM43). Their smooth derivatives on the compact closed square prove membership in the actual $H$. Explicit compact majorants follow by differentiating $\rho e^{A^o_\lambda\rho^2}$: they are $\rho e^{A^o_\lambda\rho^2}$, $(1+2A^o_\lambda\rho^2)e^{A^o_\lambda\rho^2}$, and $(6A^o_\lambda\rho+4(A^o_\lambda)^2\rho^3)e^{A^o_\lambda\rho^2}$ for orders zero, one, and two; use the actual $\rho=1/4$ and the exponential factor as in (GM40).