Will signal get distorted if channel bandwidth is lesser than the signal bandwidth?
The key here is the title of the book you are reading: "digital communicatinon by Taub schilling " as you say

What the book is talking about is digital communications and not analog communications in which the fidelity of the transmitted signal is of great importance; we want the output signal to be as close a replica of the input signal as is possible. In digital communications, it does not matter diddly-squat if the transmitted waveform is distorted as long as from the received waveform one can determine which of the two (more generally, few) possible transmitted waveforms $s_0(t)$ and $s_1(t)$ caused this received signal to occur. If the receiver can make this determination with high accuracy, then we have managed to transmit one bit (or more generally $\log_2$(few bits)) with high accuracy by choosing $s_0(t)$ or $s_1(t)$ according as we wish to send a $0$ or a $1$. Even if the received signal
$r(t)$ is badly distorted from what was sent, the receiver can still determine whether the transmitter sent $s_0(t)$ or $s_1(t)$: fidelity of transmission is unimportant.

With the above in mind, note that the Taub and Schilling description is dreadfully ambiguous, and in many spots downright incorrect. An equalizer (a concept from digital communications) cannot perfectly undo the distortion caused by forcing a 1 kHz analog signal through a low-pass filter with 3 dB bandwidth $1$ Hz. An inverse (analog) filter is trivial to write down --- it is a filter whose transfer function is $\frac{1}{H(f)}$ where $H(f)$ is the channel response --- but in most cases that inverse filter is unrealizable. The truth of the matter is that any filtering by a channel causes the transmitted signal to be spread out in time and reversing this time spreading and bringing the signal back to its pristine shape is in almost all cases impossible: once the genie is out of the lamp, he cannot be stuffed back in, ditto Pandora's box for those preferring a more Western analogy. That last sentence "The signal is then recoverable precisely as transmitted" is sheer nonsense.