Given a time-dependent unbounded linear operator

$A={\left({A}_{t}\right)}_{t\ge 0}$ on a separable Hilbert space

$H$, generating a semigroup

${\left({U}_{s,t}\right)}_{s\ge t\ge 0}\subset L\left(H\right)$ which is strongly continuous in

$(s,t)$, and given a cylindrical Brownian motion

$W$ defined over some probability space

$({\Omega},\mathcal{F},P),$ the author of the present paper investigates the

$H$-valued backward stochastic differential equation (BSDE)

$d{Y}_{t}=-\{{A}_{t}{Y}_{t}+f(t,{Y}_{t},{Z}_{t})\}dt+{Z}_{t}d{W}_{t},\phantom{\rule{0.166667em}{0ex}}t\in [0,T],\phantom{\rule{0.166667em}{0ex}}{Y}_{T}=\xi \in {L}^{2}({\Omega},{\mathcal{F}}_{T}^{W},P;H)$. Supposing that the

$\left({\mathcal{F}}_{t}^{W}\right)$-progressively measurable coefficient

$f(t,\omega ,y,z)$ is Lipschitz in

$z$ and such that

$|f(t,y,z)-f(t,{y}^{\text{'}},z)|\le c(|y-{y}^{\text{'}}\left|\right),\phantom{\rule{0.166667em}{0ex}}y,{y}^{\text{'}}\in H,$ for some non increasing concave function

$c:{R}_{+}^{*}\to {R}_{+}^{*}$ with

$c(0+)=0$ and

${\int}_{{0}^{+}}^{1}{c}^{-1}\left(t\right)dt=+\infty ,$ the author shows the existence and the uniqueness for this BSDE. Afterwards he proves that the process

$Y$ has continuous trajectories. The author’s work concerns a subject which enjoys a great interest since the pioneering paper by

*Y. Hu* and

*S. Peng* [Stochastic Anal. Appl. 9, No. 4, 445–459 (1991;

Zbl 0736.60051)], and a lot of generalizations and applications of the equation considered by Hu and Peng (the above BSDE with time-independent unbounded linear operator

$A$) have been studied since then. The present paper employs standard methods for its generalization.