Odd and even singular-value transforms

Let $A:H_R\to H_L$ be the PUE map of Projected-unitary encoding and singular vectors, with positive singular-value decomposition $A={\sum }_{j=1}^r{\sigma }_j\mathrm{\mid }{u}_j⟩⟨v_j\mathrm{\mid }$. Let $p:[-1,1]\to \mathbb{C}$ be a scalar function.

If $p$ is odd, define the odd singular-value transform by

$p_{\mathrm{o}\mathrm{d}\mathrm{d}}^{\mathrm{S}\mathrm{V}}(A):H_R\to H_L,p_{\mathrm{o}\mathrm{d}\mathrm{d}}^{\mathrm{S}\mathrm{V}}(A)={\sum }_{j=1}^{r}p({\sigma }_j)\mathrm{\mid }{u}_j⟩⟨v_j\mathrm{\mid }\mathrm{.}$

If $p$ is even, define its right-space transform by

$p_R^{\mathrm{S}\mathrm{V}}(A):H_R\to H_R,p_R^{\mathrm{S}\mathrm{V}}(A)={\sum }_{j=1}^{r}p({\sigma }_j)\mathrm{\mid }{v}_j⟩⟨v_j\mathrm{\mid }+p(0){\mathrm{\Pi }}_{\ker A},$

where ${\mathrm{\Pi }}_{\ker A}$ is the orthogonal projection in $H_R$ onto $\ker A$. The left-space version is

$p_L^{\mathrm{S}\mathrm{V}}(A):H_L\to H_L,p_L^{\mathrm{S}\mathrm{V}}(A)={\sum }_{j=1}^{r}p({\sigma }_j)\mathrm{\mid }{u}_j⟩⟨u_j\mathrm{\mid }+p(0){\mathrm{\Pi }}_{\ker A^{\ast }}\mathrm{.}$

The odd transform annihilates the right kernel because oddness gives $p(0)=0$; the even transforms use the canonical kernel projections. The resulting operators are independent of the chosen SVD by Singular-value transform is independent of SVD choices ↗.