if breakdown does not occur and each matrix $T_n+\sigma_\ell I_n$is nonsingular, then $\bigr\|{\mbox{\boldmath $r$}}_n^{(\ell), {\textrm}{\scriptsize{SQ($B$)}}}\bigl\|_2=\bigr\|{\mbox{\boldmath $r$}}_n^{(\ell), {\textrm}{\scriptsize{SCOCG}}}\bigl\|_2\ge \bigr\|{\mbox{\boldmath $r$}}_n^{(\ell), {\textrm}{\scriptsize{SQ}}}\bigl\|_2 \ {\textrm}{for}\   \ell=1,2,\ldots m$, where the superscripts SQ($B$), SCOCG, and SQ are short for shifted QMR\_SYM($B$), shifted COCG, and shifted QMR\_SYM, respectively;

-   $ \|{\mbox{\boldmath $r$}}_n^{(\ell), {\textrm}{\scriptsize{SQ($B$)}}}\|_2=|\widetilde{g}_{n+1}^{(\ell)}|\  {\textrm}{for} \  \ell=1,2,\ldots,m, \  n\ge 0.$

[*Proof*]{}. The proof of ($I$) is the same as that of Proposition 2.2, and is based on the fact that under the assumption the complex symmetric Lanczos process generates real basis vectors.

Next, we give a proof of ($I\hspace{-0.5mm}I$). The $n$th residuals of the shifted COCG method [@Shifted_COCG] belong to ${\mbox{\boldmath $b$}}-(A+\sigma_\ell I)K_n(A+\sigma_\ell I,{\mbox{\boldmath $b$}})$. Hence, each ${\mbox{\boldmath $r$}}_n^{(\ell), {\textrm}{\scriptsize{SCOCG}}}$ can be written as ${\mbox{\boldmath $r$}}_n^{(\ell), {\textrm}{\scriptsize{SCOCG}}}={\mbox{\boldmath $b$}}-(A+\sigma_\ell I)V_n{\mbox{\boldmath $y$}}_n^{(\ell), {\textrm}{\scriptsize{SCOCG}}}$, where $V_n$ is the same matrix as in (\[comp\_sym\_Lanczos\_Mat\]) Since each ${\mbox{\boldmath $r$}}_n^{(\ell), {\textrm}{\scriptsize{SCOCG}}}$ is orthogonal to each subspace $K_n(\overline{A}+\overline{\sigma}_\ell I,\overline{{\mbox{\boldmath $b$}}})${=$K_n(\overline{A},\overline{{\mbox{\boldmath $b$}}})\}$, i.e., ${\mbox{\boldmath $r$}}_n^{(\ell), {\textrm}{\scriptsize{SCOCG}}} \perp K_n(\overline{A},\overline{{\mbox{\boldmath $b$}}})$, we have $V_n^{\textrm{\scriptsize{T}}}{\mbox{\boldmath $b$}}-V_n^{\textrm{\scriptsize{T}}}(A+\sigma_\ell I)V_n{\mbox{\boldmath $y$}}^{(\ell), {\textrm}{\scriptsize{SCOCG}}}=0$, and thus it follows from (\[comp\_sym\_Lanczos\_Mat\]) that we have the relation ${\mbox{\boldmath $y$}}_n^{(\ell), {\textrm}{\scriptsize{SCOCG}}} = \{V_n^{\textrm{\scriptsize{T}}}(A+\sigma_\ell I)V_n\}^{-1}V_n^{\textrm{\scriptsize{T}}}{\mbox{\boldmath $b$}}= g_1(T_n^{(\ell)})^{-1}{\mbox{\boldmath $e$}}_1$, where $T_n^{(\ell)}:=T_n+\sigma_\ell I_n$. Since the shifted QMR\_SYM($B$) method has the form (\[QMR\_residual\]), it is sufficient to show that ${\mbox{\boldmath 