some important question to understand partial least squares (PLS)

Question

I am reading the PLS of Kevin Dunn's book Process Improvement Using Data section 6.7. Introduction to Projection to Latent Structures (PLS) I don't understand following things of NIPALS algorithm page 383:

1. If we ignore the later Deflation part, could I understand $X_{N\times n},Y_{N\times m}$ can be decomposited as (up to k-th component) $$X = U_kA_k + E_X = u_1W_1 + \cdots + u_kW_k + E_X;$$ $$Y = T_kB_k + E_X = t_1C_1 + \cdots + t_kC_k + E_Y;$$ here $U_k = ( u_1, \cdots, u_k), T_k = ( t_1, \cdots, t_k)$ (actually unlike the NIPALS of PCA, I am not sure which loading and score vector correspond $X$ and $Y$) And the regression $Y = X\beta_{n\times m}$ is equivalent to the regression between $T_k = U_k\gamma_{k\times k},$ which can be regard as the simple linear regression of each column of $T_k$ as all components of $T_k/U_k$ are uncorrelated.

   Am I right up to now? Does that mean the number of components at most $min(n,m)?$

2. If I understand correctly, then go forward to the next Deflation part. I don't understand the wording in page 383:

   And there’s the problem: the values in Y𝑎 are not available when the PLS model is being used in the future, on new data. In the future we will only have the new values of X.

   $X,Y$ are the given training sample matrixs and $u_i,t_i,W_i,C_i$ are all known after the training. Why can't we do as step 1. predict the new $X^*$ by $Y^* = X^*\beta_{n\times m}?$

3. In the calculation of loading vector $p_a$ corresponding the score vector $t_a,$ It is a regression of matrix (y) onto a vector(x). But from the result $\dfrac{X'_at_a}{t'_at_a},$ it seems still calculated as a vector onto a matrix like the normal linear regression. Do you know why?

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