Let \(T= (t_{nk})\) be a row-finite matrix such that each column belongs to the space \(bv\) of sequences of bounded variation. The \(T\)-sections of a sequence \(x= (x_ k)\) are defined as \(t^ n x:= \sum_ k t_{nk} x_ k e^ k\), where \(e^ k:= (\delta_{ki})_ i\) \((k\in \mathbb{K})\). With respect to a locally convex sequence space \(E\), a sequence \(x\) is said to have property UTK (unconditional \(T\)-sectional convergence), UFTK or UTB if the net \((\sum_{n\in F}(t^ n x- t^{n-1} x))_{{\mathcal F}}\) converges to \(x\in E\), is weak Cauchy in \(E\) or is bounded in \(E\), respectively [see \textit{D. J. Fleming}, Math. Z. 194, 405-414 (1987; Zbl 0596.40006)]. (Here \(\mathcal F\) denotes the collection of all finite subsets of \(\mathbb{N}\).) Let \(E_{\text{UTK}}\), \(E_{\text{UFTK}}\), \(E_{\text{UTB}}\) denote the space of all \(x\) which have respectively UTK, UFTK, UTB in \(E\). The space \(E\) is said to have property UTB if \(E_{\text{UTB}}\supset E\supset \varphi:= \text{limspan}\{e^ k\mid k\in \mathbb{N}\}\), and similarly for the other properties. For \(E\) containing \(\varphi\), define \(E_{\text{AD}}:= \overline\varphi\), \(E^ f:= \{(f(e^ k))\mid f\in E'\}\) and \(E^{\alpha_ T}:= \{u= (u_ k)\mid ux:= (u_ k x_ k)\in bv_ T\) \((x\in E)\}\), where \(bv_ T:= \{x= (x_ k)\mid Tx:= (\sum_ k t_{nk} x_ k)_ n\in bv\}\). \(E\) is called a sum space if \(E^ f= M(E,E):= \{u=(u_ k)\mid ux\in E\) \((x\in E)\}\). The author introduces and investigates the notion of \(T\)-solidness for sequence spaces. \(E\) is called \(T\)-solid if \(E\supset bv^ f_ T\cdot E:= \{ux\mid u\in bv^ f_ T\), \(x\in E\}\); cf. \textit{D. J. Fleming} [op. cit.]. It is shown that any \(T\)-solid FK-space \(E\) is UFTK and \(E^{\alpha_ T}= E^ f\). If \(bv_ T\) is a sum space, then for any locally convex sequence space \(E\) containing \(\varphi\) the space \(E_{\text{UTB}}= E_{\text{UFTK}}\) is \(T\)-solid. If in addition \(T\) is \(bv\)-reversible, then \(E_{\text{UTK}}\) is \(T\)-solid. The author extends the notions of the solid hull and the solid kernel of an FK-space [see \textit{J. M. Anderson} and \textit{A. L. Shields}, Trans. Am. Math. Soc. 224 (1976), 255-265 (1977; Zbl 0352.30032)] to \(T\)-solidness. \textit{M. Buntinas} and \textit{N. Tanović-Miller} [Proc. Am. Math. Soc. 111, No. 4, 967-979 (1991; Zbl 0771.46006)] have introduced the notions of absolute sectional boundedness \(|\text{AB}|\) and absolute sectional convergence \(|\text{AK}|\), and they have showed that an FK-space is solid if and only if it is \(|\text{AB}|\). The author gives natural extensions to \(T\)-sectional properties. A sequence \(x\) is said to have property \(|\text{TB}|\) in \(E\) if the set \(\{(\sum_{n\in\mathbb{N}} (t_{nk}- t_{n-1,k})x_ k\mid N\subset\mathbb{N}\}\) is bounded. It is said to have property \(|\text{TK}|\) if in addition the nets \(((\sum_{n\in F\cap N}(t_{nk}- t_{n-1,k})x_ k)_ k)_{F\in {\mathcal F}}\) converge to \(((\sum_{n\in N} (t_{nk}- t_{n-1,k})x_ k)_ k)_{F\in {\mathcal F}}\) in \(E\) uniformly for \(N\in \mathbb{N}\). Let \(T\) be \(bv\)-reversible. Then \(E\) is \(|\text{TB}|\) if and only if it is \(T\)-solid. If in addition \(bv_ T\) is a sum space, then the following assertions are equivalent: (i) \(E\) is \(|\text{TK}|\), (ii) \(E\) is UTK, (iii) \(E\) is \(T\)-solid and AD, (iv) \(E= p_ 0\cdot E\), where \(p_ 0:= (bv^ f_ T)_{\text{AD}}\).