Basic question about automorphic representations

Question

I'm reading Gelfand/Graev/Pyatetskii-Shapiro's book Representation Theory and Automorphic Functions. On page 20, the setting is:

Let $G$ be a locally compact topological group, $\Gamma\le G$ a discrete subgroup (assumed to be cocompact).

Let $V$ be a finite dimensional vector space (which I assume they want to be over $\mathbb{C}$), and $\chi : \Gamma\rightarrow V$ a unitary representation of $\Gamma$ (so I guess they want $V$ to be equipped with an inner product). Let $H(\chi)$ denote the Hilbert space of measurable functions $f : G\rightarrow V$ satisfying:

1. $f(\gamma g) = \chi(\gamma)f(g)$ for all $\gamma\in\Gamma,g\in G$.
2. $(f,f) := \int_{\Gamma\backslash G}[f,f]dx < \infty$, where $[f,f]$ denotes the inner product of $V$.

Here, I think they really want the integral to be over a fundamental domain, and I think the choice of fundamental domain doesn't matter because $\chi$ is unitary.

For $g\in G$, they define the operator $T(g)$ on the space $H(\chi)$ by $T(g)f(x) := f(xg)$. One can check that $T : g\mapsto T(g)$ is a unitary representation on $H(\chi)$.

Let $\varphi$ be a continuous function on $G$ of compact support. Then they also define an operator $T_\varphi$, which I assume is an operator on $H(\chi)$, given by $$T_\varphi := \int\varphi(g)T(g)dg$$ where I assume the integral is over $G$, and for $f\in H(\chi)$, they want $$(T_\varphi f)(x) := \int_G\varphi(g)f(xg)dg$$ For two continuous compactly supported functions $\varphi_1,\varphi_2$ on $G$, let $\varphi_1*\varphi_2$ denote the convolution. On page 21, they claim:

1. $T_{\varphi_1*\varphi_2} = T_{\varphi_1}T_{\varphi_2}$ (I assume $T_{\varphi_1}T_{\varphi_2}$ denotes composition of operators?)
2. $T_{\varphi^*} = T_\varphi^*$, where $\varphi^*(g) = \overline{\varphi(g^{-1})}$ (presumably the bar denotes complex conjugation?)
3. "$T_{\varphi*\varphi*}$ is a self-adjoint positive definite operator."

I don't understand what they mean by (3). Do they mean $T_{(\varphi *\varphi)^*}$? I'm also having some trouble verifying that (1) and (2) are true. (also I can't check (3), since I don't know what they mean).

My field is normally algebraic geometry, and I haven't worked with integrals for many years now, so perhaps this is all relatively easy, but I would appreciate it greatly if someone could spell out how to check these three properties.

EDIT: Also, given two subspaces $H_1,H_2$ of a hilbert space $H$, what does $H_1\stackrel{.}{+} H_2$ mean?