Invariant Factorization of LPDOs - Transpose

Transpose

Factorization of an operator is the first step on the way of solving corresponding equation. But for solution we need right factors and BK-factorization constructs left factors which are easy to construct. On the other hand, the existence of a certain right factor of a LPDO is equivalent to the existence of a corresponding left factor of the transpose of that operator.

Definition The transpose of an operator \mathcal{A}=\sum a_{\alpha}\partial^{\alpha},\qquad \partial^{\alpha}=\partial_1^{\alpha_1}\cdots\partial_n^{\alpha_n}. is defined as \mathcal{A}^t u = \sum (-1)^{|\alpha|}\partial^\alpha(a_\alpha u). and the identity \partial^\gamma(uv)=\sum \binom\gamma\alpha \partial^\alpha u,\partial^{\gamma-\alpha}v implies that \mathcal{A}^t=\sum (-1)^{|\alpha+\beta|}\binom{\alpha+\beta}\alpha (\partial^\beta a_{\alpha+\beta})\partial^\alpha.

Now the coefficients are

\tilde{a}_{\alpha}=\sum (-1)^{|\alpha+\beta|} \binom{\alpha+\beta}{\alpha}\partial^\beta(a_{\alpha+\beta}).

with a standard convention for binomial coefficients in several variables (see Binomial coefficient), e.g. in two variables

\binom\alpha\beta=\binom{(\alpha_1,\alpha_2)}{(\beta_1,\beta_2)}=\binom{\alpha_1}{\beta_1}\,\binom{\alpha_2}{\beta_2}.

In particular, for the operator the coefficients are \tilde{a}_{jk}=a_{jk},\quad j+k=2; \tilde{a}_{10}=-a_{10}+2\partial_x a_{20}+\partial_y a_{11}, \tilde{a}_{01}=-a_{01}+\partial_x a_{11}+2\partial_y a_{02},

\tilde{a}_{00}=a_{00}-\partial_x a_{10}-\partial_y a_{01}+\partial_x^2 a_{20}+\partial_x \partial_x a_{11}+\partial_y^2 a_{02}.

For instance, the operator

is factorizable as

and its transpose is factorizable then as

Read more about this topic:  Invariant Factorization Of LPDOs

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